The Hidden Correlation of Collateralized Debt Obligations

Transcription

1 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

2 Acknowledgments I would like to thank my supervisor Dr Alonso Peña for 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 engagement 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

3 Abstract We propose a model for the correlation structure of reference portfolios of collateralized 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

4 Contents List of Figures List of Tables v vi 1 Introduction to Collateralized Debt Obligations The CDO Market Valuation of STCDOs Outline Modeling of Multivariate Default Risk Structural Models Copula Functions Factor Models Default Correlation Implied Correlation Correlation Smile Modeling the Correlation Matrix Empirical Correlations and RMT Group Models Attainable Correlation Smiles Numerical Implementation Integration Schemes Existence and Uniqueness of Correlation Performance Conclusions 51 iv

5 CONTENTS CONTENTS A Inverse Transform Sampling 53 B Correlation in Student-t Factor Model 54 C Used Parameters 56 Bibliography 59 v

6 List of Figures 3.1 Example of correlation smile/skew observed in the market Loss distribution, un-skewed vs. skewed Example of implied correlations for non-flat correlation matrix Implied corr. for varying µ in the two-layer hierarchical model Implied corr. for varying in the two-layer hierarchical model Implied corr. for varying n in the two-layer hierarchical model Implied corr. for varying ν in the two-layer hierarchical model Implied corr. for varying n in the two-layer hierarchical model Implied corr. for varying µ ρ in the two-layer hierarchical model Implied corr. for varying ρ in the two-layer hierarchical model Implied corr. for varying n in the block model Implied corr. for varying ρ in the block model Integration schemes for the 1-factor Gaussian copula Integration schemes for the 2-factor Student-t copula Integration schemes for the 2-factor Gaussian copula Uniqueness and existence of implied compound correlations vi

7 List of Tables 4.1 Types of group models Test cases for the validation of the developed program code Tranche spreads: LHP vs. numerical results in factor model framework CPU times for CDO pricing in different parameter regimes C.1 Used parameters for the validation of integration schemes C.2 Used parameters for the implied correlations of group models vii

8 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 obligations (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 billion 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

9 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 1 An arbitrage CDO can also be set up as cash CDO. 2

10 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: CDO 2 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 CDO n ). Callable STCDOs are the equivalent to bermudan options in equity markets and are particularly difficult to price due to their loss path dependence (cp. footnote 2 in Chapter 2). The focus of this work lies on standard CDOs, though. With respect to financial stability, the CDO market was believed to have a positive impact, as it facilitates the distribution of credit risk to non-bank investors and increases the degree of completeness of the credit market [12]. However, the narrowing of the corporate spreads in the years before the financial crisis (which, due to the leverage, had a multiplicative effect on the CDO spreads) increased the pressure to find higher yield/higher risk underlyings to keep engagement into CDOs profitable for investors and arrangers, bringing subprime mortgage loans into focus. However, this selective choice of loans also implied a higher correlation than anticipated from historical estimates: the subprime loans are more correlated because they are all underwritten with weak credit standards. This was not adequately reflected by both the credit rating agencies, which continued to assign first-class ratings to the senior CDO tranches, as well as by the market participants who continued to invest. As default rates on mortgage loans started to rise in 2007, the affected CDOs experienced severe losses and rating downgrades. For the case of synthetic CDOs, systemic risk also emerges from the fact that the number of liquid CDS names is limited and arrangers are forced to constantly draw from the same pool of names. Consequently, the same name is contained in many CDO deals, greatly amplifying the impact of a default. Also, an important point to consider is the back-action of the growth of the CDO issuance on other markets: as pointed out above, the arranger of a STCDO needs 3

11 1.2 Valuation of STCDOs 1 Introduction to Collateralized Debt Obligations to hedge the credit risk he is exposed to, namely by selling protection on the CDS market, thereby contributing to the narrowing of the spreads. Due to the leverage of the STCDOs, this has to be done with a multiple of the notional value of the sold tranches. Moreover, these developments were probably supported by the high complexity of the involved products, with models and techniques that were not always sufficiently tested and understood. Especially on the investor s side, the lack of expert knowledge probably led to CDO investments without the ability to correctly assess and manage the associated risk. 1.2 Valuation of STCDOs As pointed out above, the investor of a STCDO covers only the tranche losses of the tranche he is invested in, i.e. the losses in the interval [a L, a H ], where a L (a H ) is referred to as lower (upper) attachment point and the size (or notional) of the tranche is defined as a H a L. Each loss within this tranche results in a payment equal to the loss amount from the investor to the arranger; in analogy to a credit default swap, this is called the protection leg of the STCDO. As compensation, the investor receives periodic payments (premium leg), usually paid quarterly in arrears 2, at a spread s on an outstanding principal which is initially equal to the notional of the tranche and stochastically declines with each loss by the size of the loss. Based on this, the STCDO can be valued as a swap contract, with the value from the investor s point of view equal to the expected present value of the premium payments less the expected present value of the protection payments. The expected tranche loss at time t reads E [L tr (t)] = E [ (min(l(t), a H ) a L ) +], (1.1) where the expectation is taken with respect to the risk neutral measure and L(t) is the portfolio loss, see Eq. (2.1). If the loss distribution F (x; t) P(L(t) x) is known (e.g. in the LHP approximation, see Sec ), the expected tranche loss can be written as E [L tr (t)] = I a L (min(x, a H ) a L )df (x; t), (1.2) where I is the maximal loss, i.e. the initial portfolio amount. 2 Except for the equity tranche, where a part of the premium is usually paid upfront. 4

12 1 Introduction to Collateralized Debt Obligations 1.3 Outline For a portfolio of finite size, we use the (numerically obtained) discrete loss probabilities P (w; t) P(L(t) = wu) (see Sec for the meaning of the loss unit u), E [L tr (t)] = w P (w; t) (min(wu, a H ) a L ) +. (1.3) With this, the value of the premium leg evaluates to V premium = s n ( D(t i ) t i ah a L E [L tr (t i )] ), (1.4) i=1 where s is the spread, 0 < t 1 <... < t n = T are the payment dates, D(t i ) is the discount factor for time t i, and t i t i t i 1 is the accrual factor. The value of the protection leg reads [ T V protection = E 0 ] D(t ) dl tr (t ), (1.5) however, one commonly assumes that the payments are made on the next periodic payment date after the default (otherwise, one would also have to compute the loss distribution for each banking day in between the payment dates), thereby essentially neglecting the interest that the payment would accrue since the default, V protection = n D(t i ) (E [L tr (t i )] E [L tr (t i 1 )]). (1.6) i=1 The fair (or par) spread s Par is the spread that sets the total value of the tranche to zero, s Par = V protection i D(t ( ). (1.7) i) t i ah a L E [L tr (t i )] Hence, central for the valuation of the STCDO is the determination of the portfolio loss distribution F (x; t), respectively P (w; t). This will be the main focus of Chapter Outline From the discussions in Sec. 1.1 above, it should have become apparent that the correlation among the names in the reference portfolio of a CDO is of crucial importance. It is therefore the primary topic of this thesis. In Chapter 2, we will cover the basics of the modeling of multivariate default risk in order to derive the loss distribution. It will become clear how the correlation 5

13 1.3 Outline 1 Introduction to Collateralized Debt Obligations between the names in the underlying portfolio of the CDO enters into the valuation. The Gaussian copula framework with flat correlation is introduced as the standard market model for the valuation of CDOs. The different concepts of historical vs. implied correlation are pointed out in Chapter 3, with the emphasis on the two methodologies for implied correlation, namely compound and base correlation. The properties, advantages, and pitfalls of these two measures are discussed. The starting point for the investigations in this thesis is the empirical observation that the market reports a non-constant correlation across the different tranches of a CDO (called smile, or skew, respectively) that contradicts the assumption of a constant correlation in the Gaussian copula framework. In order to reproduce the correlation smile, a model for the correlation matrix is proposed in Chapter 4, based on the investigation of empirical correlation matrices of stock price returns and on economic intuition. The shapes of the implied correlation curves emerging from this model are analyzed in detail for different parameter regimes. Chapter 5 finally focuses on some numerical issues that one encounters when implementing a CDO valuation code. 6

14 Chapter 2 Modeling of Multivariate Default Risk The aim of this chapter is the derivation of the portfolio loss distribution F (x; t) in Eq. (1.2) respectively the discrete loss probabilities 1 P (w; t) from Eq. (1.3). To this end, we employ different techniques from default risk modeling like structural models, copulas, and factor models. Let us introduce some notation: we consider a reference portfolio of N obligors subject to default risk with random default times τ 1, τ 2,..., τ N. Modeling of multivariate default risk denotes the modeling of the joint distribution H(t 1, t 2,...) P(τ 1 t 1, τ 2 t 2,...) with marginal CDFs p i (t) P(τ i t) = E [1 τi t]. p i (t) is called the default probability of name i and is assumed to be known, calibrated from credit curves (e.g. risky bonds or credit default swaps) on name i. Thus, one only needs to specify the dependency function (copula) for a full characterization of H, see Sec The default of obligor i results in a loss l i = N i (1 R i ) (for recovery type CDOs), where N i is the notional of the i-th credit and R i [0, 1] is its recovery rate, i.e. the percentage of the notional that is expected to be recovered in case of default by selling assets of the obligor. The random portfolio loss is then given by L(t) = N l i 1 τi t, (2.1) i=1 1 We will henceforth use the term loss distribution also for P (w; t). 2 To be more precise, we point out that the valuation of a CDO doesn t require knowledge about the full joint distribution of default times which also contains all inter-temporal correlations, but rather the portfolio loss distribution which is the marginal distribution of losses at a fixed point in time. As described in Ref. [5], for the CDO valuation only the loss distribution at the discrete set of payment dates is needed. 7

15 2.1 Structural Models 2 Modeling of Multivariate Default Risk with expectation value E [L(t)] = N t i=1 0 E [l i τ i = s] dp i (s). (2.2) Note that no default dependence enters here, i.e. the expected loss over the whole portfolio (respectively tranche structure) depends only on the single-name default probabilities. The expected loss suffered by part of the structure like a tranche is significantly affected by the default dependence between the obligors, though. This makes the modeling of the default dependence in Sec. 2.2 a key issue for the valuation and risk management of CDOs. In a first step, we cover the modeling of univariate default risk, though. The factor model approach that is followed in Sec. 2.3 derives from Merton s structural model approach [33] for univariate default risk, we will therefore introduce structural models in the following Structural Models Structural models (also termed asset value models or firm value models) define the time of default of a firm as the moment when the firm s assets fall below its liabilities. The asset value is modeled as random process X t and looking at a fixed time scale (maturity) t the default is given by τ t X t (t) C, (2.3) where the barrier C stands for the liabilities. The Black-Cox approach [8] extends this model by allowing a default anytime before maturity at the first hitting time of a (possibly time varying) barrier, τ = inf{t : X t (t) C(t)}. (2.4) We will focus on the simplified case of static models that suppress the time dependence of X by treating it as random variable; let C be an increasing function of time. This relates the random variables X and τ by X = C(τ). 3 We will not cover intensity-based models [25] in this work. We will, however, sometimes use the notion of the hazard rate h, borrowed from the intensity framework. For our purposes, the default probability and the hazard rate are related by p(t) = 1 e ht. 8

16 2 Modeling of Multivariate Default Risk 2.2 Copula Functions In order to calibrate the static model to the market implied default curve p(t), the level of the barrier is adjusted. We have p(t) = P(X C(t)) = P(F X (X) F X (C(t))) = P(U F X (C(t))) = F X (C(t)), (2.5) where F X is the distribution of X, and U denotes a uniformly distributed random variable (we have used the fact that F X (X) is uniformly distributed, cp. Appendix A). The calibration equation for C(t) then reads C(t) = F 1 X (p(t)). (2.6) Structural models naturally extend to the modeling of multiple dependent defaults since a default dependence can easily be incorporated by a correlation between the individual asset processes (variables, respectively). This will be useful in the context of factor models below. 2.2 Copula Functions The specification of a joint distribution of default times is central to the valuation of basket credit derivatives. It is economically plausible and empirically shown that default events of obligors are not independent from each other but exhibit positive dependence, i.e. due to economic cycles or firm interactions, defaults tend to cluster together. Consequently, the joint default probability cannot simply be modeled by the product of the individual default probabilities. Obviously, the marginal single-name default probabilities discussed in Sec. 2.1 don t uniquely fix the joint distribution. A convenient and powerful approach to specify the missing part, i.e. the default dependence between the obligors, is the use of copula functions. In general, copulas allow for the separate treatment of marginal distributions and dependency structure by linking given marginal distributions to a joint distribution. In this section a very brief and informal introduction to copulas is given, see e.g. Ref. [14] for further reading and proofs. A copula is defined as a multivariate distribution function on the unit hypercube with uniformly distributed marginals, C(u 1,..., u n ) = P(U 1 u 1,..., U n u n ), C(1,..., 1, u i, 1,..., 1) = u i, u i [0, 1]. (2.7) The idea of the analysis of dependency with copula functions is expressed by Sklar s theorem [36]: if F (x 1,..., x n ) : R n [0, 1] is a n-dimensional distribution function 9

17 2.2 Copula Functions 2 Modeling of Multivariate Default Risk with marginal distributions F 1 (x 1 ),..., F n (x n ), then there exists a copula function C (C is unique if the F i (x i ) are all continuous) such that for all (x 1,..., x n ) R n F (x 1,..., x n ) = C(F 1 (x 1 ),..., F n (x n )) (2.8) holds, i.e. C joins the marginal distributions to the joint distribution. In practice, if a joint distribution function with a suitable dependency structure has been identified, the copula containing the dependency structure can be extracted by C(u 1,..., u n ) = F (F 1 1 (u 1 ),..., F 1 n (u n )). (2.9) Then, a set of given marginal distributions (e.g. the individual credit curves p i (t) obtained from the market) is plugged into Eq. (2.8), resulting in a joint distribution function with the given marginal distributions and the extracted dependency structure. For the popular multivariate Gaussian, the extracted Gaussian copula reads (Φ being the standard normal CDF) C ρ (u 1,..., u n ) = Φ ρ (Φ 1 (u 1 ),..., Φ 1 (u n )) = 1 (2π) n/2 detρ Φ 1 (u 1 )... Φ 1 (u n) ( exp 1 ) 2 xt ρ 1 x dx. (2.10) For the special case of the Gaussian copula, the dependency structure is fully described by the matrix of pairwise correlations ρ. The (linear) correlation between two random variables X and Y with nonzero finite variances σ(x) and σ(y ) is defined as ρ(x, Y ) = E[XY ] E[X]E[Y ]. (2.11) σ(x) σ(y ) Linear correlation is measure of linear dependence, since Y = ax + b almost surely iff ρ(x, Y ) = 1. For joint distributions other than the multivariate Gaussian, the correlation does not completely define the dependency structure 4. Other measures of dependency have been proposed; however, since the Gaussian copula is the industry standard in credit risk modeling, the correlation is commonly used as measure of dependency, despite its limited applicability. Due to its particular importance for the valuation of CDOs, correlation is also used as quotation device, see Chapter 3. 4 This even applies to other distributions belonging to the class of elliptical distribution, like the Student-t distribution, see Appendix B. 10

18 2 Modeling of Multivariate Default Risk 2.3 Factor Models 2.3 Factor Models Factor models are a widely used simplification in the context of credit risk modeling and provide an efficient way to deal with a large number of credits with default dependence [18]. We assume a low- (d-) dimensional random variable the factor Z G Z conditional upon which default times become independent, i.e. the Z- conditional defaults are assumed independent events with probability P(τ t Z = z) p i (t Z). (2.12) The independence of the conditional default probabilities allows for the application of recursion arguments that lead to large gains in the efficiency of the calculation of the (discrete) conditional loss probabilities P (w; t Z) P(L(t) = wu Z), see Sec The unconditional loss distribution then follows from integrating P (w; t Z) over the distribution G Z of Z, P (w; t) = Ω P (w; t Z = z) dg Z (z). (2.13) The integration domain Ω depends on the properties of the random variable(s) that are taken as factor(s), e.g. Ω = R d for a d-dimensional Gaussian factor structure, see Sec , and Ω = R d R + for a d-dimensional Student-t factor structure, see Sec The form of the conditional default probability p i (t Z) : R + Ω [0, 1] emerges from the used factor model with the constraint that the (unconditional) default probabilities of the individual firms are preserved, p i (t Z = z) dg Z (z) =! p i (t). (2.14) Ω Following the structural model approach and notation, the random variable X describing the asset value in a factor model is given by 5 X i = β i Z + 1 βi 2 ɛ i. (2.15) The systematic factor Z (which can be thought of as a common macroeconomic driver or, loosely speaking, the market ) is shared by all X i, and ɛ i is the residual driver idiosyncratic to firm i and is independent of Z and ɛ j factor loadings. j i. The β s are called In the following, we describe two common models: the Gaussian factor model and the Student-t factor model. 5 The products in general need to be taken as scalar products, β i Z β i Z, β i β j βi 2 β i β i 11 β i β i β j,

19 2.3 Factor Models 2 Modeling of Multivariate Default Risk Gaussian Copula The Gaussian copula model [28] is the industry standard model. For a Gaussian copula, Z and ɛ are independent N (0, 1) distributed. Being a weighted sum of two standard Gaussian random variables, X i is also standard Gaussian and the correlation ρ between two obligors directly evaluates to ρ(x i, X j ) = { βi β j, for i j 1, for i = j (2.16) i.e. X = (X 1,..., X N ) is a N-dimensional Gaussian variable with correlation matrix ρ. With Φ the standard normal distribution, the conditional default probability reads ( ) C i (t) β i Z p i (t Z) = P(X i < C i (t) Z) = Φ. (2.17) 1 β 2 i Since X i N (0, 1) and following Eq. (2.6) the calibration equation for C i (t) reads C i (t) = Φ 1 (p i (t)). (2.18) Student-t Copula The Student-t distribution has fatter tails than the Gaussian distribution and is known to generate tail dependence in the joint distribution, two properties that are also observed in the market. We make use of the following model: X i = β i ν ν ν g V + 1 βi 2 g V i = g ( ) β i V + 1 βi 2 V i, (2.19) where V and V i are independent N (0, 1), and g is an independent χ 2 (ν) random vari- able that is common to both the systematic process Z = ν V and the idiosyncratic g process ɛ i = ν g V i. In order to prevent infinite low-order moments of X, we impose the restriction that ν > 2. Note that the factor Z is now two-dimensional and has a bivariate distribution func- tion. The asset variable X i, though, reads X i = ν Y g i with Y i N (0, 1) which amounts to a Student-t distribution with ν degrees of freedom, X i t ν, as desired. Also, recall that with Y being a N-dimensional standard Gaussian variable with correlation matrix ρ, and g a scalar χ 2 (ν) random variable, X = ν g Y follows a 12

20 2 Modeling of Multivariate Default Risk 2.3 Factor Models N-dimensional Student-t distribution 6 with correlation matrix ρ 7 and ν degrees of freedom. Thus, we have in fact constructed a Student-t copula. For the conditional default probability we now obtain ( g ν p i (t V, g) = Φ C ) i(t) β i V, (2.20) 1 β 2 i where the calibration equation reads C i (t) = t 1 ν (p i (t)). Here, the strength of the model chosen above becomes apparent: if we had chosen two independent χ 2 distributed variables for the systematic and the idiosyncratic factor, then X i would have been a sum of two Student-t variables whose distribution cannot be expressed in closed form. The calibration of C would therefore have required a numerical root finding in contrast to the analytic expression above. On the downside, the model above requires a double integral in Eq. (2.13) over both factor components g and V The Portfolio Loss Distribution From the conditional default probabilities, the conditional loss distribution P (w; t Z) can be calculated. Let us first consider the case of a homogeneous setup, i.e. l i l, p i p, and β i β are identical for all obligors. It follows that the loss distribution (conditional and unconditional) is related to the distribution of the number K of defaulted names by the transformation P (w; t) = P(K = w/l; t), i.e. we only have to consider the density of the number of defaults. Given the conditional independence of defaults, the density is given by the binomial density function, ( ) N P(K = n; t Z) = p(t Z) n (1 p(t Z)) N n = b(n; N, p(t Z)). (2.21) n In the more general inhomogeneous setup, the conditional loss distribution is given by the convolution product of all conditional single name loss distributions 8. This convolution can be computed by established methods for the calculation of continuous 6 The generalization of the univariate Student-t distribution to a multivariate distribution is not unique. We use the definition used in Ref. [7]. 7 Hence, the construction in Eq preserves the Gaussian correlation structure, i.e. as before we get ρ(x i, X j ) = β i β j, for the derivation see Appendix B. This means that the factor loadings β have the same meaning as before and form a decomposition of the correlation matrix. 8 The distribution of the sum X + Y of two independent random variables with respective distributions G X (x) and G Y (x) is given by the convolution G (x) = x = G X(x)G Y (x x). The distribution of X X n, X i independent, can be computed from this by induction. 13

21 2.3 Factor Models 2 Modeling of Multivariate Default Risk convolutions of independent random variables, see Ref. [20] for a Fourier transform based method. A particularly simple and efficient method [7] is based on recursion: a loss unit u is introduced, such that the individual losses l i are discretized in integer units w i of u, l i = w i u 9. If L n is the loss associated with a portfolio containing the first n names (in arbitrary order), then P n (w; t Z) P(L n (t) = wu Z) is recursively given by P n+1 (w; t Z) = P n (w w n+1 ; t Z) p n+1 (t Z) + P n (w; t Z)(1 p n+1 (t Z)). (2.22) From this, the loss distribution P (w; t Z) = P N (w; t Z) can be built, starting from the boundary case of the empty portfolio, P 0 (w; t Z) = δ w,0. Note that Eq. (2.22) corresponds to a nested loop: new names are successively added to the portfolio and for each new name the probabilities for all possible losses w = 0,..., w max,n + w n+1 need to be computed. Therefore, the computational time of numerical implementations will typically depend roughly quadratically on the size of the portfolio (cp. Sec. 5.3) Fitting the Factor Loadings Suppose we have inferred a correlation matrix ρ for the reference portfolio from the market, e.g. from asset correlations (cp. Chapter 3). In general, the factor loadings cannot be chosen such that the correlation matrix is exactly reproduced, since ρ has (N 1)(N 2)/2 independent entries whereas there are only N factor loadings for each factor. Therefore, one aims to find factor loadings β that approximate ρ best with a given number of factors M. This optimization problem can be numerically solved with a number of methods, an iterative algorithm based on principal component analysis (PCA) is sketched in Ref. [7]. From the spectral decomposition of the correlation matrix, ρ = V ΛV T, only the eigenvectors of the M largest eigenvalues are used, β 1 = λ 1 v 1,..., β M = λ 1 v M (where the eigenvalues are labeled in decreasing order of magnitude), yielding an approximation ρ to ρ, ρ = M i=1 β i ( βi ) T. In general, the factor loadings β i are not optimal yet, since the optimization procedure above minimizes tr {( ρ ρ )( ρ ρ ) T }, i.e. it uses a Frobenius norm on all elements of ρ ρ. However, the diagonal elements of ρ don t need to be approximated, since the 9 See Ref. [7] for a discussion about the optimal size of the loss unit in order to keep rounding errors at the one hand and computational effort at the other hand at a tolerable level. 14

22 2 Modeling of Multivariate Default Risk 2.3 Factor Models construction of the factor models (2.15) assures that ρ(x i, X i ) = 1, and we only need to consider non-diagonal elements of ρ. Therefore, a modified correlation matrix { ρ (2) ρij, for i j ij = (2.23) ρ ij, for i = j is approximated in the next iteration, yielding β (2) i, etc. In this thesis we will follow a different approach, though, as we will construct factor loadings that build a model correlation matrix with some predefined structure, see Chapter Large Homogeneous Portfolio Approximation For an asymptotically large, homogeneous portfolio it is possible to derive a closedform solution for the loss distribution. Specifically, we make the following assumptions: The portfolio consists of an infinite number of entities N. The notional amounts, single name default probabilities, recovery rates, and pair-wise correlations are identical for all entities. There is a single factor determining the conditional default probabilities. This is referred to as the Large Homogeneous Portfolio (LHP) approximation. We now consider the average number of defaulted obligors K and use the fact that conditional N on the factor Z defaults are independent and follow the law of large numbers, ( ) K P N = p(z) Z = 1, (2.24) i.e. the average number of defaulted obligors equals the conditional default probability almost surely. As above, we obtain the unconditional (now continuous) distribution by integrating out the factor, ( ) K F (α) = P N α = = ( ) K P N α Z = z dg Z (z) P(p(Z) α Z = z) dg Z (z). (2.25) p(z) is not a random variable but a deterministic function and we can restrict the integration domain to the region where p(z) α, i.e. where z p 1 (α) (for p(z) is a decreasing function), F (α) = p 1 (α) dg Z (z) = 1 G Z (p 1 (α)). (2.26) 15

23 2.3 Factor Models 2 Modeling of Multivariate Default Risk Gaussian Copula For the Gaussian copula, Eq. (2.26) evaluates to ( C ) ( ) 1 β F (α) = 1 Φ 2 Φ 1 (α) 1 β2 Φ 1 (α) C = Φ, (2.27) β β where we have used the symmetry of the Gaussian density function in the last step. The density function reads f(α) = d dα F (α) = g Z(p 1 1 (α)) p (Z(α)) ( ) 1 β2 Φ 1 (α) C 1 β 2 = ϕ β βϕ ( ), (2.28) Φ 1 (α) where ϕ is the density of the standard normal distribution Student-t Copula As pointed out above, the conditional default probability (2.20) in the Student-t copula model differs from the Gaussian one by being conditioned on two independent variables, thereby failing the conditions for the LHP approximation stated above. This issue is resolved in Ref. [35] by introducing a new factor variable η, g η := C βv, (2.29) ν thus transforming Eq. (2.20) into ( ) η p(η) = Φ. (2.30) 1 β 2 Eq. (2.26) then turns into F (α) = p 1 (α) since p(η) is an increasing function. dg η (η) = G η (p 1 (α)), (2.31) It is shown in Ref. [35], that the density function g η can be written in a closed form as finite sum over incomplete gamma functions 10, g η (t) = A e νt 2 2(C 2 +νβ 2 ) 10 Assuming that ν N and p < 50% [35]. ν 1 k=0 ( ) ν 1 (Bt) ν 1 k I k k ( Bt), (2.32) 16

24 2 Modeling of Multivariate Default Risk 2.3 Factor Models where for each k N 0 the function I k : R R is given by { 2 k 1 2 Γ ( ) ( ) k+1 I k (x) = 2 Γ k+1, x2, for x I k (0) + ( 1) k (I k (0) I k ( x)), for x < 0 (2.33) with Γ(k, x) = 1 Γ(k) x A := e t t k 1 dt, and the constants A and B are defined as ( ) ν β ν 1 ν 2 C 2 +νβ 2 ν 1 π 2 2 Γ ( ), B := ν 2 C β C 2 + νβ 2. (2.34) This can be used to express the density function of the fraction of defaulted obligors as f(α) = g η (p 1 1 (α)) p (η(α)) ( ) = g η 1 β2 Φ 1 1 β (α) 2 ϕ ( ). (2.35) Φ 1 (α) 17

25 Chapter 3 Default Correlation As pointed out in the previous chapters, the valuation of CDO tranches involves a number of parameters: the reference portfolio is described by single-name credit spreads, recovery rates, and losses, as well as the correlation between the obligors; for the CDO being a cash-flow instrument, its valuation naturally also requires risk-free interest rates and accrual factors. The correlation, though, stands out from this list as it is the only parameter specific to CDOs (or other basket credit derivatives), whereas the others can be observed from the liquid CDS and interest rate markets. Empirical investigations suggest that defaults tend to cluster together and that there is positive correlation even between companies in different sectors; therefore, the assumption of uncorrelated defaults is not justified and correlation has to be considered. Moreover, the strength of the correlation crucially affects the prices and risk profiles of the CDO tranches. The determination of the correlation therefore receives particular interest and can be approached from two opposite directions. One can consider default correlation as model input that needs to be estimated externally, to the best of one s knowledge. This is complicated by the fact that there is little empirical data on the relationship between defaults, and the underlying process driving the defaults is not easily observable. In theory, information about default correlation should be extracted from historical defaults; this generally suffers from the low number of observed defaults, though. For the estimation of the default correlation one therefore typically resorts to the correlation of historical equity market returns which is used as proxy for the asset return correlation of latent variable models like the Merton model (2.3). 18

26 3 Default Correlation 3.1 Implied Correlation 3.1 Implied Correlation Conversely, one can think of correlation as a quotation convention. With the advent of standardized STCDOs on liquid CDS indices and the resulting increased liquidity in the CDO market, prices can be expected to follow from supply and demand rather than from a theoretical model. The correlation implied from these prices can be used as a quotation device to facilitate a comparison of prices across tranches and products, the line of thought being similar to implied volatilities of equity markets 1. Given the developments in the standardization of the credit derivative markets, there is also a need for a commonly agreed method of quoting this implied correlation. As standard model, the one-factor Gaussian copula with a flat correlation, β i β, ρ ij ρ i, j is widely used. However, like the Block-Scholes model in equity markets, the one-factor Gaussian copula is known to be too simplistic to reflect market reality. It can therefore not be expected that a single correlation number is sufficient to fit the prices of all tranches simultaneously, but rather that each tranche trades at its own implied correlation. In the following, we describe two popular methods that have been developed for quoting the implied correlation with respect to the tranche Compound Correlation Compound correlation is the direct extension of the Black-Scholes implied volatility and was the quotation convention initially used in the market. By inverting the standard pricing model, it is possible to find the level of default correlation that equates the observed market spread and the theoretical spread for a tranche. The (compound) correlation affects the different tranches in different ways. In general, an increase in default correlation means that it becomes more likely to observe many or few defaults. An equity tranche, for instance, is not much affected by the occurrence of many defaults as it absorbs losses only up to its detachment point. However, the occurrence of only a few defaults reduces the expected loss in the tranche and therefore the fair spread. Thus, in terms of prices, an equity tranche investor is long correlation. Conversely, if defaults in the portfolio are completely uncorrelated, the probability of the occurrence of many losses is very low, i.e. the losses are very unlikely to hit the senior tranche. Therefore, senior tranche investors are short correlation. Mezzanine tranches, finally, are in general not monotonic in (compound) correlation: a long position in a mezzanine tranche with attachment point a L and detachment 1 However, market prices may be observed that are not attainable by any choice of correlation. 19

27 3.1 Implied Correlation 3 Default Correlation point a H can be seen as a long position in an equity tranche with detachment point a H plus a short position in an equity tranche with detachment point a L, since the expected loss can be decomposed into E [L al,a H ] = E [L 0,aH ] E [L 0,aL ]. (3.1) Since both virtual tranches are long correlation but enter with opposite sign, the root finding algorithm for the determination of the compound correlation often doesn t yield a unique compound correlation but two correlations that produce the same tranche PV, see Sec. 5.2 for an example. This ambiguity is one of the biggest drawbacks of compound correlation. Moreover, as we will discuss in Sec. 3.2, compound correlation is not easy to extend to the pricing of tranches with non-standard strikes. The upside of compound correlation is its transparency and intuitive meaning as it directly maps to the asset correlation of latent variable models Base Correlation Base correlation has been developed by JP Morgan [32] and makes use of the monotonicity of equity tranches. The basic idea of base correlation is the decomposition of all tranches into combinations of (virtual, non-traded) base (=equity) tranches, as given in Eq. (3.1). In contrast to compound correlation, where the two virtual base tranches are priced at the same correlation, base correlation allows for pricing each base tranche at a different correlation. This is done via a bootstrapping mechanism where the base correlation from the first tranche is used to solve for the base correlation of the second one, and so on: 1. Let V a b (s, ρ) denote the PV (from the investor s perspective) of the tranche with attachment point a and detachment point b, with spread s, and (flat) correlation ρ. The base correlation of the equity tranche, ρ 0 K1, solves the equation V 0 K1 (s 0 K1, ρ 0 K1 ) =! 0, (3.2) where s 0 K1 is the market spread for the 0 K 1 tranche (i.e. the equity tranche). This is equivalent to the determination of the compound correlation for the equity tranche; hence, base correlation and compound correlation are identical for the equity tranche. 20

28 3 Default Correlation 3.1 Implied Correlation 2. The next tranche K 1 K 2 is equivalent to a portfolio of a long position in a 0 K 2 base tranche and a short position in a 0 K 1 base tranche, both using the K 1 K 2 market spread. We define the base correlation ρ 0 K2 as the correlation that solves! V K1 K 2 (s K1 K 2, ρ K1 K 2 ) = V }{{} 0 K2 (s K1 K 2, ρ 0 K2 ) V 0 K1 (s K1 K 2, ρ 0 K1 ). }{{} =0 by definition of ρ K1 K 2 known, <0 since s K1 K 2 <s 0 K1 (3.3) 3. The procedure continues in the same way for the higher tranches. Since the PV of the 0 K 2 base tranche, V 0 K2 (s K1 K 2, ρ 0 K2 ), is the only changing component in Eq. (3.3) (the base correlation of the lower strike, ρ 0 K1, is fixed during the root finding) and is an increasing function of the base correlation ρ 0 K2, the base correlation is guaranteed to be unique. Moreover, by means of Eq. (3.3), the existence of base correlations for all tranches guarantees that Eq. (3.1) holds, which is a basic arbitrage-free requirement. With respect to the whole tranche structure this means that the sum of the protection legs of all tranches equals the sum of the protection legs of all underlying CDS. This is also the reason that sometimes base correlation fails to find a solution for the super senior tranche: since the expected loss (and hence the spread) over the whole tranche structure doesn t depend on the correlation (cp. Eq. (2.2)), a market spread that only slightly deviates from this spread cannot be reproduced anymore by any base correlation value. However, there are some pitfalls with base correlation, too, the most obvious being the difficulty to build intuition about it. Since every tranche is actually characterized by two correlations (one for the lower, one for the upper strike), it is not obvious how base correlation changes with default or asset correlation. In fact, an increased default correlation can result in the decrease of the base correlation for some tranches [37]. One also needs to keep in mind that the determination of the base correlation for a particular tranche requires the knowledge about the prices for all more junior tranches and that base correlation is only unique given the set of attachment points [37]. The base correlation of the a b tranche thus depends not only on a and b but also on the position of all prior attachment points, making the comparison e.g. between the European and American markets difficult due to the different tranche structure. 21

29 3.2 Correlation Smile 3 Default Correlation 3.2 Correlation Smile As pointed out above, the implied correlation is typically not constant throughout the tranches but is known to exhibit a smile (in terms of compound correlation) or a skew (in terms of base correlation), reflecting the difference between real world pricing and the assumptions of the (simplistic) Gaussian copula model, see Fig The main features of the smile/skew are relatively high spreads for equity and senior tranches (resulting in a low implied compound correlation for equity tranches and a high correlation for senior tranches) and low spreads for mezzanine tranches. Figure 3.1: Example of a correlation smile/skew observed in the market (DJ itraxx, 13 April 2006). Figure was taken from Ref. [15]. The reasons for the shape of the smile/skew are manifold, including model effects and supply-demand conditions. On the model side, it is obvious that the single correlation number isn t able to capture the dependency structure of a portfolio of 125 credits; also, the Gaussian copula model does not reflect the empirical joint distribution of default times as it underestimates the probability of a very low or very high number of defaults. On the supply-demand side it is worth noting that mezzanine tranches are very popular among investors, resulting in a tightening of spreads. On the other hand, the CDO market is segmented, i.e. different parts of the seniority structure are driven by different investors (hedge funds buying equity tranches, retail investors looking for mezzanine tranches, and pension funds focusing on senior tranches), and there are only few market participants that seek to exploit perceived mispricing across different tranches. A number of other effects have been suggested as explanation, see e.g. Refs. [4, 32]. 22

30 3 Default Correlation 3.2 Correlation Smile An important application of the correlation smile/skew is the interpolation in order to calculate spreads for customized tranches on standard indices with non-standard attachment points. In this context, the shape of the base correlation skew, which is typically close to linear, appears to be much better suited for an interpolation than the more complicated smile shape of compound correlation. However, as it is shown in Ref. [37], the spread errors from a linear interpolation of the base correlation skew can still be large. In rare cases, the interpolation can even yield negative expected losses for some tranches Modeling Approaches A starting point for modeling approaches is the loss distribution implied by the market. Since tranche prices are essentially integrals over the loss distribution, cp. Eqs. (1.2) and (1.3), the implied loss distribution can be backed out from the market by differentiating. Compared to the loss distribution generated from the Gaussian copula, one observes a fat upper tail, indicating an increased probability of high loss scenarios (yielding a high spread for senior tranches), and a lower probability of zero (or few) defaults (equivalent to a high spread for the equity tranche), see Fig Whereas the fat upper tail can be modeled e.g. by switching from the Gaussian to a Student-t copula, the challenge is to do so without also raising the probability of small losses. 50% 100% Density function un-skewed skewed Prob P(L<x) un-skewed skewed 0% Normalised Loss 0% Normalised Loss Figure 3.2: Distribution of losses, as produced by a Gaussian copula ( un-skewed ) and observed in the market ( skewed ). Figure was taken from Ref. [24]. A number of alternative copulas have been proposed to overcome this issue, including Double-t, Clayton, Archimedean, and Marshall-Olkin copulas, we refer to Ref. [11] for an overview. Also extensions to the Gaussian copula model have been investigated: stochastic correlation interprets the correlation as (binary) stochastic variable 23