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 the credit-market counterparts to the familiar CMO structure.
Introduction
• Since their introduction in mid 1990s, the
market for CDOs has grown dramatically. • CDO market now in excess of $1.5 trillion. • 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. • First large scale empirical analysis of how CDOs are priced in the market.
Introduction
• Motivated by Duffie and Garleanu (2001),
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.
Introduction
• 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 2003-2005 period.
Overview
• 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 second jump has interpretation of a joint default of firms in a sector or industry. • 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).
Overview
• Probability of a firm defaulting can be
partitioned into three events: only that firm defaults, the firm and others in the industry default together, and 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.
Overview
• We test for how many factors are actually
needed to price CDOs. • All three are needed. • Direct evidence that defaults cluster and are not independent.
Overview
• How well does the model fit? • 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. 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.
Literature
• 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), 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. • 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.
Introduction to CDOs
• Remaining tranches are typically 7-10, 1015, 15-30, and 30-100 tranches, where the first number is the attachment point. • Collectively, tranches represent entire capital structure of a synthetic bank. • Each tranche has its own credit rating. • Even if no AAA bonds in markets, could synthesize AAA 30-100 super senior debt.
Introduction to CDOs
• Synthetic CDO structures replace the underlying • • •
portfolio of bonds with a basket of credit default swaps. Simpler, cash flows are easier to define. 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.
The CDX Index
• 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. • Reconstituted every six months. CDX4 included Ford and GM, CDX 5 dropped them because they were no longer investment grade.
Summary Statistics
Tranche Spreads
Correlation of Spread Changes
The Model
• Total portfolio losses • Portfolio losses
• Intensity dynamics
• Conditional on path, probability of i jumps
is
• Let
• This function satisfies PDE
• Solution is
Modeling the Index
• Premium leg
• Protection leg
Modeling Tranches
• Tranche losses • Premium leg • Protection leg
Tests for Number of Factors
Parameter Estimates
Estimated Intensities
Percentage of Spread
CDX Index Tranche Pricing Errors
Is Dispersion Priced?
Conclusion
• 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. Direct evidence of market expectations about default clustering. Identifies the idiosyncratic and common components of default risk.