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					An Empirical Analysis of the
Pricing of Collateralized Debt

     Francis Longstaff, UCLA
     Arvind Rajan, Citigroup

• 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.
• 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.
• Important drivers of growth include the
  creation of standardized CDX and ITraxx
  indexes. Also, the parallel growth of the
  credit derivatives market.

• 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.
• Motivated by evidence in the literature that
    credit spreads are driven by multiple factors,
    we first develop a three-factor portfolio credit
•   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)).

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

• 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.
• 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
•   RMSE was small even during May 2005 credit
• 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.
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
Introduction to CDOs

• Remaining tranches are typically 7-10, 10-
  15, 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
•   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.
The Model

• Total portfolio losses

• Portfolio losses
• Intensity dynamics
• Conditional on path, probability of i jumps

• Let
• This function satisfies PDE
• Solution is
Modeling the Index

• Premium leg

• Protection leg
Modeling Tranches

• Tranche losses

• Premium leg

• Protection leg
• A portfolio credit model explains virtually all the
    time series and cross sectional variation in CDO
•   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.