Credit Risk Models IV: Understanding and pricing CDOs Abel Elizalde ∗ CEMFI and UPNA December 2005 † Abstract Some investors in the Collateralized Debt Obligations (CDOs) market have been publicly accused of not fully understanding the risks and dynamics of these products. They won’t have an excuse any more. This report explains the mechanics of CDOs: their implied cash flows, the variables affecting those cash flows, their pricing, the sensitivity of CDO prices to those variables, the functioning of the markets where they are traded, their different types, the conventionsusedfortradingCDOs,... WebuiltourdescriptionofCDOspricing upon the Vasicek asymptotic single factor model because of its simplicity and theinsightsitprovidesregardingthepricingofCDOs. Additionally,weprovide an extensive and updated review of the literature which extends the Vasicek model by relaxing its, somehow restrictive, assumptions in order to build more realistic and, as a consequence, more complicated CDO pricing models. CEMFI, Casado del Alisal 5, 28014 Madrid, Spain. Email: [email protected]. ∗ First version: July 2005. An updated version is available at www.abelelizalde.com. † Contents 1 Introduction 1 2 Structural model for credit risk: Merton (1974) 4 3 Vasicek asymptotic single factor model 6 4 CDOs 11 4.1 Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.3 Vasicek model: homogeneous large portfolio . . . . . . . . . . . . . . 17 4.4 Sensitivity of tranche premiums to the model parameters . . . . . . . 18 4.5 Trading issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.5.1 Moral hazard . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.5.2 Types of CDOs . . . . . . . . . . . . . . . . . . . . . . . . . . 22 4.5.3 Correlation smile and base correlations . . . . . . . . . . . . . 23 4.5.4 Exotic CDOs . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.6 Extensions of the Vasicek model . . . . . . . . . . . . . . . . . . . . . 26 4.6.1 Homogeneous finite portfolio . . . . . . . . . . . . . . . . . . . 27 4.6.2 General distribution functions . . . . . . . . . . . . . . . . . . 28 4.6.3 Heterogeneous finite portfolio . . . . . . . . . . . . . . . . . . 29 4.6.4 Stochastic default correlations . . . . . . . . . . . . . . . . . . 31 4.6.5 Multifactor models . . . . . . . . . . . . . . . . . . . . . . . . 32 4.6.6 Random loss given default . . . . . . . . . . . . . . . . . . . . 32 4.6.7 Totally external defaults . . . . . . . . . . . . . . . . . . . . . 32 4.6.8 Focusing on a single tranche . . . . . . . . . . . . . . . . . . . 33 4.7 Parameter calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.7.1 Default probabilities . . . . . . . . . . . . . . . . . . . . . . . 34 4.7.2 Loss given default . . . . . . . . . . . . . . . . . . . . . . . . . 34 4.7.3 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Appendix 38 A Bank capital regulation . . . . . . . . . . . . . . . . . . . . . . . . . . 38 References 41 This paper is part of a series of surveys on credit risk modelling and pricing. The complete list of surveys is available at www.abelelizalde.com, and consists on the following: 1. Credit Risk Models I: Default Correlation in Intensity Models. 2. Credit Risk Models II: Structural Models. 3. Credit Risk Models III: Reconciliation Reduced-Structural Models. 4. Credit Risk Models IV: Understanding and pricing CDOs. 5. Credit Default Swap Valuation: An Application to Spanish Firms. “there is a minority of investors - perhaps 10 per cent - who do not fully understand what they are getting into.” Michael Gibson, head of trading risk analysis at the US Federal Reserve (Financial Times, 2005a). “Understanding the credit risk profile of CDO tranches poses challenges even to the most sophisticated participants.” Alan Greenspan, chairman of the US Federal Reserve (Financial Times, 2005b). The “sharp increase in the complexity of credit derivative products being traded in the past couple of years ... may also mean that investors do not fully understand what they are purchasing in areas such as collateralised debt obligations (CDOs) - or pools of debt linked securities.” (Financial Times, 2005c). “Last month, Bank of America and Italian bank Banca Popolare di Intra (BPI)settledtheir40millioneurolawsuit, inwhichBPIclaimsitwasmis- sold several CDO investments by Bank of America. ... It would be naive to think that this is the last court case that will emerge. A number of investors and regulators have already voiced concern about the level of complexity in some investment products. With something as complicated as CDO-squared, it’s not hard to imagine more investors claiming they were mis-sold investments if the credit cycle takes a turn for the worse.” Nick Sawyer, Editor (Risk, 2005). 1 Introduction Imagine a pool of defaultable instruments (bonds, loans, credit default swaps CDSs, ...) from different firms is put together. The losses on the initial portfolio value due to the default of the underlying firms depend on the default probability of each firm and the losses derived from each default (losses given default). Additionally, the degree of dependence between the firms’ default probability, usually known as default correlation, plays animportant role on thetimingof the firms’ defaults (whether they tend to cluster or they are independent) and, as a consequence, on the distribution of the portfolio losses. Next, imagine we, the owners of the portfolio, decide to buy protection against the possible losses due to the defaults of the underlying firms, but we can not sell the portfolio. One way to do it is buying Credit Default Swaps (CDSs) of each firm, but that’s not the way we are interested in here. We can sell the portfolio in tranches, i.e. we can buy protection for those losses in tranches. A Collateralized Debt Obligation (CDO) consists on tranching and selling the credit risk of the underlying portfolio. For example, a tranche with attachment points [K ,K ] will bear the portfolio losses L U in excess of K percent of the initial value of the portfolio, up to a K percent. The L L tranche absorbing the first losses, called equity tranche, is characterized by K = 0 L and K > 0. The holders of a tranche characterized by attachment points [K ,K ] U L U won’t suffer any loss as long as the total portfolio loss is lower than K percent of its L initialvalue. WhenthetotalportfoliolossgoesaboveK percent, thetrancheholders L are responsible for the losses exceeding K percent, up to K percent. Losses above L U K percent of the initial portfolio value do not affect them. The lower attachment U point K of each tranche corresponds to the upper attachment point K of the L U previous (more junior) tranche. Obviously, the holders of each tranche (sellers of credit risk protection) have to be compensated for bearing those losses: they receive a periodic fee, called premium, until the maturity of the CDO (point in which they also stop being responsible for 1 future losses in the portfolio.) The premium of the equity tranche will be the highest because its holders absorb the first losses of the portfolio. In order for the holders of more senior tranches to start suffering losses, the holders of more junior tranches would have already born all losses they were exposed to (K K percent of the U L − initial portfolio value). As a consequence, the higher the seniority of the tranche the lower the premiums holders receive. The whole problem lies in determining the tranches’ premiums. They have to compensate tranche holders for the expected losses they will suffer and, therefore, they depend on the distribution of the portfolio losses which, as we argued above, depends on the underlying firms’ default probabilities, default correlations, and losses given default. Our review of CDO pricing models focus on a particular branch of this litera- ture: the ones based on structural models. The main distinguishing characteristic of such models with respect to the other credit risk modelling alternative, reduced form models, is the link they provide between the probability of default and the firms’ fundamental financial variables: assets and liabilities. The way structural models incorporate the dependence between the firms’ default probabilities (which is a key ingredient for CDO pricing) is by making such fundamental variables depend on a set of, generally unobserved, common factors. In contrast, reduced form models rely on market prices of the firms’ defaultable instruments to extract both their default probabilities and their credit risk dependen- cies. These models rely on the market as the only source of information regarding the firms’ credit structure and do not consider any information coming from their balance sheets. Although easier to calibrate, reduced form models lack the link be- tween credit risk and the firms’ financial situation incorporated in their assets and liabilities. Anyway, reduced form models provide an alternative way of pricing CDOs which one shouldn’t forget, besides their lower popularity in this area.1 In fact, these 1See,amongothers,ChavaandJarrow(2004),Driessen(2005),andElizalde(2005d)forintensity modelsincorporatingthecorrelationstructureacrossfirms,andGaliani(2003),DuffieandGarleanu 2 modelsprovide thedynamics needed to pricesome of therecent exoticCDOproducts which we review in Section 4.5.4. ThepaperstartsfromthetheoreticalfoundationsoftheVasicekmodel. Itpresents, in Section 2, a review of credit risk structural models, in order to understand the mo- tivations behind such models. Section 3 describes in detail the Vasicek asymptotic single riskfactor model, whichhas becomethe market standard for CDOpricing, and which is also referred to as the normal or Gaussian copula model, because that is the dependence structure it implies for the firms’ default correlation.2 Withallthosetoolsinhand,Section4divesintoCDOs: mechanics, types, pricing, premium sensitivity to the model parameters, trading issues (implied and base corre- lations), extensions of the Vasicek model and, finally, a few words on the calibration of the reviewed models. The assumptions of the Vasicek asymptotic single risk factor model about the characteristics of the underlying portfolio (homogeneous infinitely large portfolio, ...) simplify the analytical derivation of CDO premiums but are not very realistic. The extensions we present relax these assumptions, making the model more suitable for CDO pricing. Investment banks and rating agencies devote a large amount of effort (and money) to fine tune and improve such models in order to benefit from the increase in pricing accuracy. The text ends with an Appendix describing the application of the Vasicek factor model to bank capital regulation in Basel II, the brand new accord for banking su- pervision and regulation. Although this is not directly related with the main topic of the text, we include it because (i) it might interest some readers, (ii) it is the other most popular application of the Vasicek model, and (iii) it is straightforward using the material presented in Sections 2 and 3. Throughout the text we provide an extensive list of references which the reader (2004), and Willemann (2004) for applications of CDO pricing using intensity based methods via Monte Carlo simulation of default times. 2SeeNelsen(1999),Embrechts,McNeilandStraumann(2002),andCherubiniandLuciano(2004) for an analysis of copula functions. 3 further interested in any of the covered topics might find useful. 2 Structural model for credit risk: Merton (1974) There are two primary types of models that attempt to describe default processes in the credit risk literature: structural and reduced form models.3 Structural models use the evolution of firms’ structural variables, such as asset anddebtvalues, todeterminethetimeofdefault. Merton’smodel(1974)wasthefirst modern model of default and is considered the first structural model. In Merton’s model a firm defaults if, at the time of servicing the debt, its assets are below its out- standing debt. A second approach, within the structural framework, was introduced by Black and Cox (1976). In this approach defaults occur as soon as firm’s asset value falls below a certain threshold. In contrast to the Merton approach, default can occur at any time. Reduced form models do not consider the relationship between default and firm financial situation in an explicit manner. In contrast to structural models, the time of default in intensity models is not determined via the value of the firm, but it is the first jump of an exogenously given jump process. The parameters governing the default hazard rate are inferred from market data. Structural default models provide a link between the credit quality of a firm and the firm’s economic and financial conditions. Thus, defaults are endogenously gener- ated within the model instead of exogenously given as in the reduced approach. Merton (1974) makes use of the Black and Scholes (1973) option pricing model to value corporate liabilities. As we shall see, this is an straightforward application only if we adapt the firm’s capital structure and the default assumptions to the requirements of the Black-Scholes model. Assume that the dynamics of firm n’s asset value A follow a continuous-time n,t diffusion given, under the physical or real probability measure P, by the following 3For a literature review of credit risk models see Elizalde (2005b and c). 4 geometric Brownian motion: dA n,t = µ dt+σ dW , (1) A n n n,t n,t whereµ isthetotalexpectedreturn, σ istheasset’s(relative)instantaneousvolatil- n n ity, and W is a standard Brownian motion under P. n,t Let us assume that the capital structure of firm n is comprised by equity and by a zero-coupon bond with maturity T and face value of D . The firm’s asset value n A is simply the sum of equity and debt values. Under these assumptions, equity n,t represents a call option on the firm’s assets with maturity T and strike price of D . n It is assumed that the firm defaults if, at maturity T the firm’s asset value A is not n,T enough to pay back the face value of the debt D to bondholders. As a consequence, n the probability at time t < T of the firm defaulting at T is given by p = P[A < D A ]. (2) n,t,T n,T n n,t | This approach assumes that default can only happen at the maturity of the zero- coupon bond. It can be shown using Itô’s lemma that the diffusion process (1) allows us to express the asset value at time T as a function of the current asset value A as n,t follows σ2 A = A exp µ n (T t)+σ √T tX , (3) n,T n,t n − 2 − n − n,t,T ½µ ¶ ¾ where X is given by n,t,T W W n,T n,t X = − , (4) n,t,T √T t − and follows a standard normal distribution with zero mean and variance one.4 At time t, we can express the condition for firm n defaulting at time T in terms of the random variable X : n,t,T A < D X < K , (5) n,T n n,t,T n,t,T ⇔ 4By definition of a Brownian motion, the difference W W follows a normal distribution n,T n,t − with zero mean and standard deviation √T t. − 5 where lnD lnA µ σ2n (T t) n − n,t − n − 2 − K = . (6) n,t,T σ √³T t ´ n − As a consequence we can rewrite (2) as p = Φ(K ), (7) n,t,T n,t,T where Φ( ) is the distribution function of a standard normal random variable. · Equivalently, if instead of considering the dynamics of the asset value A under n,t the physical probability measure P, one considers its dynamics under the risk neutral probability measure Q, firm n’s risk neutral default probability is obtained. Inordertosimplifynotationhereafterwefixtheactualtimetot = 0, whichallows us to eliminate the first time subindex of p , X , and K , which become p , n,t,T n,t,T n,t,T n,T X , and K . n,T n,T 3 Vasicek asymptotic single factor model This Section builds on Vasicek (1987, 1991 and 2002).5 We are interested in the default probabilities at time t > 0 of a group of n = 1,...,N firms with the asset and liabilities structure described in the preceding section. The probability of default of each firm n at time t is denoted p and given by n,t p = Φ(K ), (8) n,t n,t lnD lnA µ σ2n t n − n,0 − n − 2 K = . (9) n,t σ √t³ ´ n Imagine we have a portfolio composed of loans to each one of the above firms (one loan per firm), and we are interested in the distribution function for the portfolio default rate, i.e. the fraction of defaulted credits in the portfolio at time t. Note that 5See also Finger (1999) and Schönbucher (2000). 6