Securitization and Copula Functions

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• Securitization and Copula Functions
• Advanced Methods of Risk Management
• Umberto Cherubini

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Learning Objectives

• In this lecture you will learn
• To evaluate basket credit derivatives using
  Marshall-Olkin distributions and copula
  functions.
• To analyze and evaluate securitization deals and
  tranches
• To evaluate the risk of tranches and design
  hedges

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Portfolios of exposures

• Assume we have a portfolio of exposures (for
  simplicity with the same LGD). We can distinguish
  between a very large number of exposures and a
  limited number of them. In a retail setting we
  are obviously interested in the former case, even
  though to set up the model we can focus on the
  latter one (around 50-100).
• We want define the probability of loss on the
  portfolio. We define Q(k) the probability of
  observing k defaults (Q(0) being survival
  probability of the portfolio). Expected loss is

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First-to-default derivatives

• Consider a credit derivative, that is a contract
  providing protection the first time that an
  element in the basket of obligsations defaults.
  Assume the protection is extended up to time T.
• The value of the derivative is
• FTD LGD v(t,T)(1 Q(0))
• Q(0) is the survival probability of all the names
  in the basket
• Q(0) ?Q(?1 gt T, ?2 gt T)

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First-x-to-default derivatives

• As an extension, consider a derivative providing
  protection on the first x defaults of the
  obligations in the basket.
• The value of the derivative will be

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Securitization deals
Senior Tranche
Originator
Junior 1 Tranche
Special Purpose Vehicle SPV
Sale of Assets
Junior 2 Tranche
Tranche
Equity Tranche
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The economic rationale

• Arbitrage (no more available) by partitioning
  the basket of exposures in a set of tranches the
  originator used to increase the overall value.
• Regulatory Arbitrage free capital from
  low-risk/low-return to high return/high risk
  investments.
• Funding diversification with respect to deposits
• Balance sheet cleaning writing down non
  performing loans and other assets from the
  balance sheet.
• Providing diversification allowing mutual funds
  to diversify investment

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Structuring securitization deals

• Securitization deal structures are based on three
  decisions
• Choice of assets (well diversified)
• Choice of number and structure of tranches
  (tranching)
• Definition of the rules by which losses on assets
  are translated into losses for each tranches
  (waterfall scheme)

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Choice of assets

• The choice of the pool of assets to be
  securitized determines the overall scenarios of
  losses.
• Actually, a CDO tranche is a set of derivatives
  written on an underlying asset which is the
  overall loss on a portfolio
• L L1 L2 Ln
• Obviously the choice of the kinds of assets, and
  their dependence structure, would have a deep
  impact on the probability distribution of losses.

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Tranche

• A tranche is a bond issued by a SPV, absorbing
  losses higher than a level La (attachment) and
  exausting principal when losses reach level Lb
  (detachment).
• The nominal value of a tranche (size) is the
  difference between Lb and La .
• Size Lb La

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Kinds of tranches

• Equity tranche is defined as La 0. Its value is
  a put option on tranches.
• v(t,T)EQmax(Lb L,0)
• A senior tranche with attachment La absorbs
  losses beyond La up to the value of the entire
  pool, 100. Its value is then
• v(t,T)(100 La) v(t,T)EQmax(L La,0)

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Arbitrage relationships

• If tranches are traded and quoted in a liquid
  market, the following no-arbitrage relationships
  must hold.
• Every intermediate tranche must be worth as the
  difference of two equity tranches
• EL(La, Lb) EL(0, Lb) EL(0,La)
• Buyng an equity tranche with detachment La and
  buyng the corresponding senior tranche
  (attachment La) amounts to buy exposure to the
  overall pool of losses.
• v(t,T)EQmax(La L,0)
• v(t,T)(100 La) v(t,T)EQmax(L La,0)
• v(t,T)100 EQ (L)

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Risk of different tranches

• Different tranches have different risk
  features. Equity tranches are more sensitive to
  idiosincratic risk, while senior tranches are
  more sensitive to systematic risk factors.
• Equity tranches used to be held by the
  originator both because it was difficult to
  place it in the market and to signal a good
  credit standing of the pool. In the recent past,
  this job has been done by private equity and
  hedge funds.

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Securitization zoology

• Cash CDO vs Synthetic CDO pools of CDS on the
  asset side, issuance of bonds on the liability
  side
• Funded CDO vs unfunded CDO CDS both on the asset
  and the liability side of the SPV
• Bespoke CDO vs standard CDO CDO on a customized
  pool of assets or exchange traded CDO on
  standardized terms
• CDO2 securitization of pools of assets including
  tranches
• Large CDO (ABS) very large pools of exposures,
  arising from leasing or mortgage deals (CMO)
• Managed vs unmanaged CDO the asset of the SPV is
  held with an asset manager who can substitute
  some of the assets in the pool.

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Synthetic CDOs
Senior Tranche
Originator
Junior 1 Tranche
Special Purpose Vehicle SPV
Protection Sale
Junior 2 Tranche
CDS Premia
Interest Payments
Tranche
Investment
Collateral AAA
Equity Tranche
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CDO2
Originator
Senior Tranche
Tranche 1,j
Junior 1 Tranche
Special Purpose Vehicle SPV
Tranche 2,j
Junior 2 Tranche
Tranche i,j
Tranche
Tranche
Equity Tranche
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Standardized CDOs

• Since June 2003 standardized securitization deals
  were introduced in the market. They are unfunded
  CDOs referred to standard set of names,
  considered representative of particular markets.
• The terms of thess contracts are also
  standardized, which makes them particularly
  liquid. They are used both to hedged bespoke
  contracts and to acquire exposure to credit.
• 125 American names (CDX) o European, Asian or
  Australian (iTraxx), pool changed every 6 months
• Standardized maturities (5, 7 e 10 anni)
• Standardized detachment
• Standardized notional (250 millions)

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  i-Traxx and CDX quotes, 5 year, September 27th
2005
 
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Gaussian copula and implied correlation

• The standard technique used in the market is
  based on Gaussian copula
• C(u1, u2,, uN) N(N 1 (u1 ), N 1 (u2 ), ,
  N 1 (uN ) ?)
• where ui is the probability of event ?i ? T and
  ?i is the default time of the i-th name.
• The correlation used is the same across all the
  correlation matrix.The value of a tranche can
  either be quoted in terms of credit spread or in
  term of the correlation figure corresponding to
  such spread. This concept is known as implied
  correlation.
• Notice that the Gaussian copula plays the same
  role as the Black and Scholes formula in option
  prices. Since equity tranches are options, the
  concept of implied correlation is only well
  defined for them. In this case, it is called base
  correlation. The market also use the term
  compound correlation for intermediate tranches,
  even though it does not have mathematical meaning
  (the function linking the price of the
  intermediate tranche to correlation is NOT
  invertible!!!)

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Monte Carlo simulationGaussian Copula

1. Cholesky decomposition A of the correlation
   matrix R
2. Simulate a set of n independent random variables
   z (z1,..., zn) from N(0,1), with N standard
   normal
3. Set x Az
4. Determine ui N(xi) with i 1,2,...,n
5. (y1,...,yn) F1-1(u1),...,Fn-1(un) where Fi
   denotes the i-th marginal distribution.

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Monte Carlo simulationStudent t Copula

1. Cholesky decomposition A of the correlation
   matrix R
2. Simulate a set of n independent random variables
   z (z1,..., zn) from N(0,1), with N standard
   normal
3. Simulate a random variable s from ?2? indipendent
   from z
4. Set x Az
5. Set x (?/s)1/2y
6. Determine ui Tv(xi) with Tv the Student t
   distribution
7. (y1,...,yn) F1-1(u1),...,Fn-1(un) where Fi
   denotes the i-th marginal distribution.

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Base correlation
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Default Probability
Correlation 0
Correlation 20
Correlation 95
MC simulation pn a basket of 100 names
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Example of iTraxx quote
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Tranche hedging

• Tranches can be hedged, by
• Taking offsetting positions in the underlying CDS
• Taking offsetting positions in other tranches
  (i.e. mezz-equity hedge)
• These hedging strategies may fail if correlation
  changes. This happened in May 2005 when
  correlation dropped to a historical low by
  causing equity and mezz to move in opposite
  directions.

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Large CDO

• Large CDO refer to securitization structures
  which are done on a large set of securities,
  which are mainly mortgages or retail credit.
• The subprime CDOs that originated the crisis in
  2007 are examples of this kind of product.
• For these products it is not possible to model
  each and every obligor and to link them by a
  copula function. What can be done is instead to
  approximate the portfolio by assuming it to be
  homogeneous .

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Gaussian factor model (Basel II)

• Assume a model in which there is a single factor
  driving all losses. The dependence structure is
  gaussian. In terms of conditional probabilility
• where M is the common factor and m is a
  particular scenario of it.

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Vasicek model

• Vasicek proposed a model in which a large number
  of obligors has similar probability of default
  and same gaussian dependence with the common
  factor M (homogeneous portfolio.
• Probability of a percentage of losses Ld

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Vasicek density function
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Vasicek model

• The mean value of the distribution is p, the
  value of default probability of each individual
• Value of equity tranche with detachment Ld is
• Equity(Ld) (Ld N(N-1(p) N-1 (Ld)sqr(1
  ?2))
• Value of the senior tranche with attachment equal
  to Ld is
• Senior(Ld) (p N(N-1(p) N-1 (Ld)sqr(1 ?2))
• where N(N-1(u) N-1 (v) ?2) is the gaussian
  copula.