dk6 coping w-copulas-managing_tl_rsk

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  • 1. CDO models: Opening the black box Part fourOctober 2008Coping with Copulas .. Managing tail risk Domenico PiconeMarco StoeckleAndrea LoddoPriya Shah +44 (0)20 7475 3870+44 (0)20 7475 3599 +44 (0)20 7475 8721 +44 (0)20 7475 6839 priya.shah@dkib.comStructured credit research: GlobalPlease refer to the Disclosure Appendix for all relevant disclosures and our disclaimer. In respect of any compendium report covering six or more companies, all relevantdisclosures are available on our website or by contacting the Dresdner Kleinwort Research Department at the addressbelow.Dresdner Bank AG London Branch, authorised by the German Federal Financial Supervisory Authority and by the Financial Services Authority; regulated by the Financial ServicesAuthority for the conduct of UK business, a Member Firm of the London Stock Exchange. Registered in England and Wales No FC007638. Located at: 30 Gresham Street, LondonEC2V 7PG. Telex: 885540 DRES BK G. Incorporated in Germany with limited liability. A Member of the Dresdner Bank Group.

2. CDO models: Opening the black box Part fourA risk management tool for CDOsAnalysing default risk of credit portfolios and CDOs In our CDO model series we have so far released various models with increasing flexibility and sophistication While they all have the advantage of being analytical / closed form solution models, they require simplifying assumptions with respect tothe way credits may default together Most importantly, they all use a factor model to generate joint default events If the way credits default together is introduced without a factor model:We cant apply the recursive algorithm anymore (as it depends on conditioning on a common factor)However, Monte Carlo (MC) techniques can be used to simulate joint default events Within a MC approach, copulas allow for a very general and flexible way to directly model the dependency within the portfolio The current credit crisis has highlighted the importance of tail events in risk management. This model gives credit investors the tool toanalyse the behaviour of their credit portfolio under stressed market conditionsCopula model: 3. Keep in mind the fine difference between correlation and dependency!!! In everyday usage, the terms correlation and dependency are often used interchangeably Linear correlation is the correlation measure we are used to (from the Markowitz portfolio theory) However, linear correlation will not be able to capture other forms of dependency correctly When we leave the Gaussian world, the linear correlation measure looses its validity and has to be handled with careSection 1Copula 101 4. CDO models: Opening the black box Part fourThe markets choiceCopulas have become the workhorse of dependency modelling Since its introduction to credit derivatives pricing by Li (1999), copulas have become the markets choice to model the dependencystructure inherent in credit portfolios The reasons for that are manifold and include ease of implementation and calibration The main benefit however is the simple and intuitive way in which even complex dependency structures can be incorporated into a genericmodel The results are dependent default times, which also can be easily calibrated to individual, market based, credit spreads This is achieved by splitting the description of the specific default behaviour of each credit in the portfolio (the marginal distributions) fromtheir joint behaviour. This means that we can model separately the individual marginals and the correlation structure (or in more generalterms their dependency structure) For simplicity, we restrict our model to cases where the marginal and the joint distribution are of the same kind, while with very little extrawork, the user can modify this aspect Copula models are powerful tools for pricing and risk management of different correlation products, ranging from FTDs to synthetic CDOtranches. In all cases the loss distributions are derived from the simulated default times, based on the assumptions used for spreads,recoveries and dependency3 5. CDO models: Opening the black box Part fourWhat is a Copula? The maths..Copulas allow expressing joint probability distributions independent from the shape of their marginals.. The joint distribution function C of m uniform random variables U1, U2, , Um can be referred to as copula function:C ( u 1 , u 2 , ... , u m , ) = Pr [U 1 u 1 , U 2 u 2 , ... , U m u m ] A set of univariate marginal distribution functions can be linked via a copula function, resulting in a multivariate distribution function: C (F1 (x 1 ), F 2 (x 2 ), ... , F m (x m )) = F (x 1 , x 2 , ... , x m ) Sklars theorem (1959) established the converse, showing that any multivariate distribution function F can be expressed via a copulafunctionFor any multivariate distribution, the univariate marginal distributions and their dependency structure can be separated Let F(x1, x2, , xm) be a joint multivariate distribution function with univariate marginal distribution functions F1(x1), F2(x2), ,Fm(xm) Then there exists a copula function C(u1,u2,,um) such that: F (x 1 , x 2 , ... , x m ) = C (F1 (x 1 ), F 2 (x 2 ), ... , F m (x m )) C is unique if each Fi is continuous Depending on the type of copula, the dependency structure will be expressed in terms of a correlation matrix, as in case for the normalcopula, or via a smaller number of parameters as in case of Archimedean copulas (such as the Clayton or Gumbel copula)4 6. CDO models: Opening the black box Part fourGenerating correlated default times - the intuition behind the Gaussian copulaCopulas to generate correlated default times400 Final step:3504First step: Translate survival rates into default300 Generate independent uniform times using the individual survival Starting with random numbers term structures250 uncorrelatedExcel =rand()200 uniforms.. 1 .0.. we finally arrive at150 10 .8correlated default100times..500 .6 00 50 100150200 250300350 4001.0 0 .430.8 0 .2 Normal Copula1 2 3 40.6 0 .0 Draws U(0,1) U(0,1) N(0,1) N(0,1) Corr N(0,1) Corr N(0,1)Corr U(0,1) Corr U(0,1) Corr DT Corr DT0.0 0 .2 0.4 0.6 0 .8 1 .0 10.07 0.86-1.46 1.07-1.46 -0.26 0.070.4019769 20.72 0.31 0.58-0.49 0.580.05 0.720.52 25490.4 30.44 0.14-0.15-1.08-0.15 -0.88 0.440.19 621250.212 210 correlated0.00.0 0.2 8 normals.. .. via correlated 6 Second step: uniforms..4 Transform uniforms in normal 2Third step:-12 -10 -8 -6-4 -2 00 2 4 6 810 12 numbers and then impose correlation structure as given by -2Map the correlated normals back to -4uniforms (Excel normsdist(z)). These the correlation matrix via -6correlated uniforms now effectively-8 Cholesky transformation (seerepresent correlated survival rates -10 next slide) to get correlated-12 normal numbers5 7. CDO models: Opening the black box Part fourDoing it in Excel..Spreadsheet screenshots - The Gaussian copula (sheet Normal and Student-t Copula)1.) Input:Choose spreads and recoveries, the level of correlation Credit 1Credit 2Correlation inputSpread (bps) 8080 70% 2 AB2 2 B = 100 % 70 % Recovery 40% 40%A2 100 % Clean Spread (bps)133.3 133.3 2.) CholeskyCorrelation Matrix transformation:100% 70% The correlation matrix is70% 100% transformed into a lower triangular matrix via theCholesky Matrix100%0% Cholesky transformation 70%71.41%3.) From uncorrelated uniform random numbers to correlated default times: three simulations for two creditsNormal CopulaDraws U(0,1)U(0,1)N(0,1)N(0,1) Corr N(0,1)Corr N(0,1) Corr U(0,1) Corr U(0,1) Corr DT Corr DT10.070.86 -1.461.07-1.46- 197 6920.720.310.58 -0.49 0.58 0.050.720.5225 4930.440.14 -0.15 -1.08-0.15-0.880.440.1962125=RAND()=NORMSINV(0.72)=0.58*100%+(-0.49)*0% =0.58*70%+(-0.49)*71.41%=NORMSDIST(0.05)Using the intensity (ie the cleanspread), the default times can bebacked out from the survivor rates via: ln (0 . 52 ) =133 . 33 bps6 8. CDO models: Opening the black box Part fourStudent-t copula..Symmetric like the Normal, but fatter tails.. The Student-t copula is a well understood alternative to the Gaussian copula While also a symmetric distribution, the Student-t distribution exhibits fatter tails (a higher kurtosis) than the normal distribution While converging towards the normal distribution for an infinite number of degrees of freedom (df), this behaviour becomes more and morepronounced for lower dfs The Student-t copula therefore is easy to implement and at the same time, it is a powerful tool to analyse the risk inherent in extremescenarios (tail events), especially when used against the Gaussian copula as a benchmark To introduce dependency via the Student-t copula we use the following transformation zt (n ) = x n Wherez ~ N (0 ,1) x ~ 2 (n ) with n denoting the degrees of freedom 7 9. CDO models: Opening the black box Part fourDoing it in Excel.. Spreadsheet screenshots - Student-t copula (sheet Normal and Student-t Copula)1.) Input:Choose spreads and recoveries, the level of correlation, and the degrees of freedom for the Student-t copulaSpread Curve Parameter InputsCredit 1Credit 2Correlation input Student-t Degrees of Freedom (df)Specify the number of degrees of freedom (df) Spread (bps) 808070% 3Recovery 40% 40% Clean Spread (bps)133.3 133.32.) Cholesky transformationIdentical to slide 63.) From uncorrelated random numbers to correlated default times:As described on the previous slide, correlated student-t random variables are generated by drawing from 2 normal distributions,correlating them and then dividing by sqrt(chi2/df).As demonstrated before for the Gaussian copula these can then be easily converted into correlated default timesStudent-t CopulaDraws U(0,1)U(0,1) N(0,1) N(0,1)Corr N(0,1) Corr N(0,1) U(0,1) Chi2 sqrt(DoF/Chi2) Corr T(


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