An Empirical Analysis of the Pricing of Collateralized Debt Obligations Francis Longstaff, UCLA Arvind Rajan, Citigroup

Introduction CDOs are financial claims to the cash flows generated by a portfolio of debt securities (or equivalently, a basket of CDS contracts). CDOs are financial claims to the cash flows generated by a portfolio of debt securities (or equivalently, a basket of CDS contracts). CDOs are the credit-market counterparts to the familiar CMO structure. CDOs are the credit-market counterparts to the familiar CMO structure.

Introduction Since their introduction in mid 1990s, the market for CDOs has grown dramatically. Since their introduction in mid 1990s, the market for CDOs has grown dramatically. CDO market now in excess of $2 trillion, with issuance in 2006 nearly doubling. CDO market now in excess of $2 trillion, with issuance in 2006 nearly doubling. Important drivers of growth include the creation of standardized CDX and ITraxx indexes. Also, the parallel growth of the credit derivatives market. Important drivers of growth include the creation of standardized CDX and ITraxx indexes. Also, the parallel growth of the credit derivatives market.

Introduction We study the pricing of CDOs using an extensive new data set recently made available to us. We study the pricing of CDOs using an extensive new data set recently made available to us. First large scale empirical analysis of how CDOs are priced in the market. First large scale empirical analysis of how CDOs are priced in the market.

Introduction Motivated by evidence in the literature that credit spreads are driven by multiple factors, we first develop a three-factor portfolio credit model. Motivated by evidence in the literature that credit spreads are driven by multiple factors, we first develop a three-factor portfolio credit model. Rather than focusing on the “quantum’’ or “zero- one’’ states of default for individual firms and then aggregating, we adopt a “statistical mechanics’’ approach and model portfolio losses directly. Also known as top-down approach (Giesecke and Goldberg (2005)). Rather than focusing on the “quantum’’ or “zero- one’’ states of default for individual firms and then aggregating, we adopt a “statistical mechanics’’ approach and model portfolio losses directly. Also known as top-down approach (Giesecke and Goldberg (2005)).

Introduction Portfolio losses are triggered by the realizations of three independent Poisson processes, each with its own intensity and jump size. Portfolio losses are triggered by the realizations of three independent Poisson processes, each with its own intensity and jump size. We take the model to the data by fitting the cross-section and time-series of CDX index tranches for the period. We take the model to the data by fitting the cross-section and time-series of CDX index tranches for the period.

Overview The implied jump sizes are 0.4, 6.0, and 35.0 percent, respectively. The implied jump sizes are 0.4, 6.0, and 35.0 percent, respectively. The first jump is.50 times 1/125 and has a clear interpretation of the idiosyncratic default of a single firm. The first jump is.50 times 1/125 and has a clear interpretation of the idiosyncratic default of a single firm. The second jump could be interpretated as joint default of firms in a sector or industry (other interpretations possible). The second jump could be interpretated as joint default of firms in a sector or industry (other interpretations possible). The third jump has interpretation of a catastrophic economywide default event. The third jump has interpretation of a catastrophic economywide default event.

Overview The expected times until a realization of these three Poisson events are 1.2, 41.5, and 763 years (under the risk-neutral measure). The expected times until a realization of these three Poisson events are 1.2, 41.5, and 763 years (under the risk-neutral measure).

Overview Probability of a firm defaulting can be partitioned into three events: that only the firm defaults, that the firm and a subset of others (industry, sector, or...) default, and that the firm and the majority of firms in the economy default together. Probability of a firm defaulting can be partitioned into three events: that only the firm defaults, that the firm and a subset of others (industry, sector, or...) default, and that the firm and the majority of firms in the economy default together. On average, these events represent 64.6, 27.1, and 8.3 percent of the credit risk of an individual firm. On average, these events represent 64.6, 27.1, and 8.3 percent of the credit risk of an individual firm.

Overview We test for how many factors are actually needed to price CDOs. We test for how many factors are actually needed to price CDOs. All three are needed. All three are needed. Direct evidence that defaults cluster and are not independent. Direct evidence that defaults cluster and are not independent.

Overview How well does the model fit? How well does the model fit? Over the 2 year period, the RMSE across CDOs is on the order of several basis points. Over the 2 year period, the RMSE across CDOs is on the order of several basis points. Initially, RMSE was higher suggesting some early pricing distortions in the market. Initially, RMSE was higher suggesting some early pricing distortions in the market. Recently, RMSE has been under one basis point. Quoted spreads can be in hundreds or even thousands. Recently, RMSE has been under one basis point. Quoted spreads can be in hundreds or even thousands. RMSE was small even during May 2005 credit crisis. RMSE was small even during May 2005 credit crisis.

Literature Many recent papers on credit derivatives and CDOs. Many recent papers on credit derivatives and CDOs. Duffie and Garleanu (2001), Bakshi, Madan, and Zhang (2004), Jorion and Zhang (2005), Longstaff, Mithal, and Neis (2005), Das, Duffie, Kapadia, and Saita (2005), Das, Freed, Geng, and Kapadia (2005), Saita (2005), Yu (2005a, b), Giesecke and Goldberg (2005), Duffie, Saita, and Wang (2006), and many others. Duffie and Garleanu (2001), Bakshi, Madan, and Zhang (2004), Jorion and Zhang (2005), Longstaff, Mithal, and Neis (2005), Das, Duffie, Kapadia, and Saita (2005), Das, Freed, Geng, and Kapadia (2005), Saita (2005), Yu (2005a, b), Giesecke and Goldberg (2005), Duffie, Saita, and Wang (2006), and many others.

Introduction to CDOs Think of a CDO as a portfolio of bonds, and tranches as claims to the cash flows from the portfolio. Think of a CDO as a portfolio of bonds, and tranches as claims to the cash flows from the portfolio. The 0-3 or equity tranche absorbs the first 3 percent of credit losses but gets highest spread of say 1500 bps on the notional. The 0-3 or equity tranche absorbs the first 3 percent of credit losses but gets highest spread of say 1500 bps on the notional. The 3-7 mezzanine tranches absorbs the next 4 percent of credit losses in return for a spread of say 300 bps on the notional. The 3-7 mezzanine tranches absorbs the next 4 percent of credit losses in return for a spread of say 300 bps on the notional.

Introduction to CDOs Remaining tranches are typically 7-10, , 15-30, and tranches, where the first number is the attachment point. Remaining tranches are typically 7-10, , 15-30, and tranches, where the first number is the attachment point. Collectively, tranches represent entire capital structure of a synthetic bank. Collectively, tranches represent entire capital structure of a synthetic bank. Each tranche has its own credit rating. Each tranche has its own credit rating. Even if no AAA bonds in markets, could synthesize AAA super senior debt. Even if no AAA bonds in markets, could synthesize AAA super senior debt.

Introduction to CDOs Synthetic CDO structures replace the underlying portfolio of bonds with a basket of credit default swaps. Synthetic CDO structures replace the underlying portfolio of bonds with a basket of credit default swaps. Simpler, cash flows are easier to define. Simpler, cash flows are easier to define. Can create single-tranche CDOs rather than having to sell entire capital structure. Can create single-tranche CDOs rather than having to sell entire capital structure. Synthetic index tranches are typically tied to a standardized index such as CDX or ITraxx. Synthetic index tranches are typically tied to a standardized index such as CDX or ITraxx.

The CDX Index The CDX investment grade North America index is an equally-weighted average of liquid CDS levels for 125 firms. The CDX investment grade North America index is an equally-weighted average of liquid CDS levels for 125 firms. Trades like a single name CDS. Trades like a single name CDS. Reconstituted every six months. CDX4 included Ford and GM, CDX 5 dropped them because they were no longer investment grade. Reconstituted every six months. CDX4 included Ford and GM, CDX 5 dropped them because they were no longer investment grade.

The Model Total portfolio losses Total portfolio losses Portfolio losses Portfolio losses

Intensity dynamics Intensity dynamics

Conditional on path, probability of i jumps is Conditional on path, probability of i jumps is Let Let

This function satisfies PDE This function satisfies PDE

Solution is Solution is

Modeling the Index Premium leg Premium leg Protection leg Protection leg

Modeling Tranches Tranche losses Tranche losses Premium leg Premium leg Protection leg Protection leg

Conclusion A portfolio credit model explains virtually all the time series and cross sectional variation in CDO prices. A portfolio credit model explains virtually all the time series and cross sectional variation in CDO prices. Market is highly efficient, even during May 2005 credit crisis. Market is highly efficient, even during May 2005 credit crisis. Direct evidence of market expectations about default clustering. Identifies the idiosyncratic and common components of default risk. Direct evidence of market expectations about default clustering. Identifies the idiosyncratic and common components of default risk.