jean-roch sibille - university of lige georges hbner university of lige third international...

Jean-Roch Sibille - University of LiègeGeorges Hübner – University of Liège

Third International Conference on Credit and Operational Risks

Pricing CDOs with Correlated Arrival and Recovery Rates


J-R. Sibille & G. Hübner

Relationship between the Probability of Default (PD) and the Recovery Rate (RR) in a Monte Carlo framework.

Pricing of tranches from Collateralized Debt Obligations (CDO).

Product sensitive to the extreme losses and to the correlation structure inside the referenced portfolio.

1. Abstract



Asset-backed Security (ABS) where a pool of fixed income instruments are repackaged into highly rated securities.

Credit Derivatives allow financial institutions to dissociate the nominal amount of a loan from the associated credit risk.

Born in the late 80s to create liquid assets for banks. The first CDO was created in 1987 by the famous Drexel

Burnham Lambert, for a $100 million loan. CBOs (CDO with Bonds) and CLOs (CDO with Loans)

2. Collateralized Debt Obligations2.1. Definition



Rapid growth of the market with a short retrenching during the short recession in 2001-2002.

Why the growth? Repackaging and transferring of credit risk You can pool whatever you want and still obtain investment grade Diversification Create liquidity Increasing loan capacity Reducing regulatory capital

Appearance of indices like ITraxx and CDX.

2.2. Growth in the Market



Growth of the CDO Market

$0,00

$20,00

$40,00

$60,00

$80,00

$100,00

$120,00

$140,00

$160,00

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006E

Year

Billi

ons



2.3. Example

Portfolio of Credit risks

(Bonds, Loans,…)

Bank

€ 1 billion

Transfer of Assets

Payment of par value of assets

SPV

Holds all the assets of

originating bankAssets

True Sale of Assets

Bank’s Balance Sheet

1st Loss Piece retainedby originator of the underlying Credits

1st Loss Piece (Equity)

2.5%

BBB Tranche Investor1.5%

A Tranche Investor1.5%

AA Tranche Investor2%

AAA Tranche Investor2.5%

Super Senior Investor

90%

Coupon payments and remaining par value at maturity

Par value cash payment

Risk

Issuance of Notes or Bonds



50 underlying zero-coupon debts with notional of 1000 monetary units.

Defaults are simulated following a Poisson process. Flat hazard rate structure of 2%. 3 tranches

Equity tranche 0% - 5% Mezzanine tranche 5% - 15% Senior tranche 15% - 100%

Maturity of 5 years. 20% default correlation across the whole portfolio. 20% correlation between RR and PD Waterfall repayment at the maturity date. 4 different hypothesis about the recovery rate 1 million scenarios generated by the simulations

2.4. CDO Prototype



2.5. Convergence



The recovery rate (RR) is the proportion of the amount due by the issuer of the debt that is recovered by the holder in case of default

Practically, three definitions are given to the RR. Recovery of face value (RFV) Recovery of Treasury value (RTV) Recovery of market value (RMV)

Here we take the RTV for simplification purposes

3. Recovery Rate3.1. Definition



A very convenient distribution for the RR is the beta distribution. The distribution is bounded between 0 and 1. There are only two parameters (alpha and beta) and they allow

for various shapes of the distribution.

3.2. RR Distribution

yxyyxxx

yxyxx

yxVarxxE

xVar

xE

32

32

2

2

:as redefined are and ,)( and )( Noting)1()(

)(

)(



Renaud and Scaillet (2004)



3.3. Correlation between RR and PD

The relationship between RR and PD has been observed on the market and different authors have build credit risk model based on this hypothesis.

The objective in this paper is to introduce this correlation in a pricing model for CDOs.



4. Pricing CDO

Use of a Gaussian copula model and Monte Carlo simulations Permits a high level of flexibility: everything can vary (default

intensity, interest rate, correlations,…). Can be very time consuming: particularly with CDOs and big baskets.



Primarily a vector of correlated random numbers is taken from a copula function.

We take the univariate cumulative probability function of these numbers and then map these probabilities with the survival function which follows a Poisson process.

Therefore we can compute the time of default for each company i and compare it to the maturity of the option to determine if a credit event has appeared

For every simulation we have the times of default of every names in the portfolio of the CDO.

Knowing the times of default we can take into account the recovery rate (RR) to determine the losses.

4.1. Basic modelnuu ,...,1

))(),...,((),...,( 111 nnn xFxFCxxF

))((1 uF



With the losses known for every simulation it is possible to construct a general probability distribution of the losses from the portfolio and realize a pricing of each tranche.

This model can be applied to a constant RR assumption or to a random RR following a beta distribution.



4.2. Correlated model The only thing we know about the state of the economy for a

specific simulation are the vector of random numbers and their corresponding time of default.

If the random numbers are generated from a Gaussian copula, it is possible to orthogonalize them with a Cholesky factorization.

Being now independent random variables, we can consider the distribution of the sum of these variables as a proxy for the state of the economy.

The variables being normal and independent, the distribution of their sum is known.



Define

We obtain the probability

Where,

n

iobsi

nobs zz

1,

)(

)(Pr( )()(

1

)( nobs

nn

i

nobsi zzZ

)Pr( )()( nobs

n zZ

),0()( nz n



The objective now is to introduce the state of the economy (SE) in the distribution of the recovery rate.

If the RR are simulated from a beta distribution, the proposition is to determine part of the RR from the cumulative distribution of SE, the other part being random.

Where,

The mean of the distribution of SE being equal to 0, it has basically no effect on the general distribution of the RR.

But this is not true because all the names do not default.

))(,,(%20),,(%80 )()( nobs

n zFxFRR

)1,0(Ux



The new objective will be to keep the correlation effect without changing the whole distribution of RR.

The solution proposed is to adapt the mean of the RR process. Normally, whatever the correlation, the average total loss on the

portfolio should be the same We can adapt the mean of the RR to the increase of the total average

loss to find back the same average loss even if the correlation effect is made.

To realize this, a new beta and alpha need to be determined for the beta distribution keeping the same standard deviation but with a new average.



4 different models (Constant, Random, Correlated and Modified Correlated RR) are tested on a virtual CDO which corresponds to market standard.

The comparison is made on the average losses that are computed for the various tranches (Equity, Mezzanine and Senior).

5. Results



A test is realized on a hundred different run of 10,000 simulations each to help determine material differences between models.

6. Significance test



Study the impact of a change in the default correlation and in the hazard rate.

7. Sensitivity Analysis



4 different hypothesis on the distribution of the RR are tested in the pricing of a CDO and significant differences appear.

Possible explanation for the correlation smile. The model needs to be confronted to market data. The calibration

is being made on the series 6 of the iTraxx. Other distribution could be tested to introduce the correlation of

default.

8. Conclusion

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