reallyisaweightedsumofsingleassetrisks,ignoringthepotentialfor diversificationeffectstypicallyinherentinaportfolio. Incontrast,ontheCDOside,itistheportfolioriskwhichendangers theperformanceofthestructure.Recallingourdiscussiononcashflow CDOs,weseethatthetranchingofnotesreallyisatranchingofthe lossdistributionofthecollateralpool,takingallpossiblediversification effectsintoaccount.Butdiversificationdecreasestheriskofaportfolio, sothatthepriceoftheportfolioriskmustbelowerthantheprice obtainedbyjustsummingupexposure-weightedsinglerisks.Thisis reflectedbythespreadsonnotesasgiveninTable8.1:Thespreads paidtonotesinvestorsaremuchlowerthanthespreadsearnedonthe bondsinthecollateralpool.Duetotherisktranchingofnotes,the spreadsonseniornotesisevenlower,duetothecreditenhancement bysubordinationprovidedfromnoteswithlowerseniority. ItisexactlythemismatchbetweenthesingleassetbasedWACofthe portfolioandthemuchlowerweightedaveragecoupononthenotesof theCDO,whichcreatesanarbitragespread.Thismismatchisinone partduetodiversificationeffects,andinanotherpartbasedonstruc- turalelementslikesubordinationorothercreditenhancementmecha- nisms.Callingspecialattentiontothediversificationpoint,onecan saythatCDOsare“correlationproducts”. Anexampleregardingarbitragespreadisgiveninthenextsection inthecontextofCDOinvestments.Conceptually,anyoriginatorofan arbitragecashflowCDOkeepingtheCDO’sfirstlosspieceautomati- callytakesontheroleoftheequityinvestor,earningtheexcessspread ofthestructureinitsownpockets.Therefore,wecanpostponethe arbitragespreadexampletothenextsection. 8.2.2TheInvestor’sPointofView VeryoftenbanksareontheinvestmentsideofaCDO.Inmanycases, ABSbondsofferinterestingandattractiveinvestmentopportunities, butrequire(duetotheircomplexity)carefulanalyticvaluationmethods forcalculatingtherisksandbenefitscomingwithanABSinvestment intothebank’sportfolio.Thiswillbemadeexplicitbymeansofthe followingexample. RecallthesamplecashflowCDOfromTable8.1.Inthisexample we assumed WAC = 10.4% and DP = 3%. Assuming an LGD of ©2003 CRC Press LLC 80%onthecollateralsecurities,weobtaintheportfolio’sexpectedloss, EL=3%×80%=2.4%. ConsideringtheCDOfromanexpectedreturnpointofview,what wouldanequityinvestorexpecttoearnonaninvestmentintheequity tranche?Atypical“back-of-the-envelope”calculationreadsasfollows: FromTable8.1weobtaintheweightedaveragecouponWAC Notes of thestructureas WAC Notes =75%×5%+10%×6.5%+5%×9.5%=4.875%, againassumingtheaverage3-monthLIBORtobeequalto4%.Because cashflowCDOscompletelyrelyonthecashflowsfromthecollateral pool,the10.4%ofthepool’sparvaluearethecompleteincomeof thestructure.Fromthisincome,allexpensesofthestructurehaveto bepaid.Paying 12 couponstonotesinvestorsyieldsagrossarbitrage spread(grossexcessspread)of [PoolIncome]−[NotesSpreads]=10.4%−4.875%=5.525%. Theexpectednetexcessspreadisthendefinedas [GrossArbitrageSpread]−EL−COSTS= =5.525%−2.4%− 450,000 300,000,000 =2.975%. Theequityreturnisthengivenby [Exp.NetExcessSpread]× PoolVolume EquityVolume =29.75%. Sothe“back-of-the-envelope”calculationpromisesaveryattractive equityreturnofalmost30%. Nowletuslookatthisseeminglyattractiveinvestmentfromaport- foliomodelingpointofview.Forthispurposewecalculatedtheequity returndistributionoftheCDObymeansofacorrelateddefaulttimes approachasoutlinedlateroninthischapter;seealsoChapter7.From aMonteCarlosimulationweobtained 13 Figure8.5.Herebyweessen- tiallyfollowedtheCDOmodelingschemeasillustratedinFigure8.3, adaptedtoadefaulttimesapproachaccordingtoFigure8.7. 12 Referring to an average scenario. 13 Under certain assumptions regarding the maturity of the bonds and the structure. ©2003 CRC Press LLC FIGURE8.5 EquityreturndistributionofaCDO. LookingattheequityreturndistributioninFigure8.5,itturnsout that,incontrasttotheaboveshown“back-of-the-envelope”calculation, theMonteCarlosimulationyieldsanaverageequityreturnofonly 15.92%.Additionally,thevolatilityofequityreturnsturnsouttobe 9.05%,sobyjustonestandarddeviationmove,theequityreturncan varybetween6.87%and24.98%.Thisreflectsthefactthatequity investmentsarerathervolatileandthereforeveryrisky.Moreover,due totaileventsofthecollateralpool’slossdistribution,itcanhappenthat thedownsiderisksofequityinvestmentsdominatetheupsidechances. Wecontinueourexamplebylookingatthereturndistributionfor class-Anotesinvestors.Table8.4showsthatin94.17%ofthecases the promised coupon of 5% has been paid to A-investors. However, in 5.83% of the cases, either not a full coupon payment or not a full repay- ment resulted in a loss. Here, loss means that at least one contractually promised dollar has not been paid. So the 5.83% are indeed the default probability of the senior tranche of the CDO. For a Aa2-rating, this is a very high chance for default. Additionally, the simulation yields an expected loss of the Aa2-tranche of 50bps, which again is very high compared to Aa2-rated bonds. Defining the loss given default of the ©2003 CRC Press LLC TABLE 8.4: Returnstatisticsforclass-A notesinvestors TABLE 8.5: Weightedaveragelifeoftranches trancheby LGD(T Aa2 )= EL(T Aa2 ) DP(T Aa2 ) = 50 583 =8.6%, showsthatontheothersidetheLGDofthetrancheisverylow.Thisis alsoduetothelargevolume(thickness)ofthetranche.InSection8.4 wewilldiscussratingagencymodels,anditwillturnoutthatagency ratingsofseniortranchestypicallyunderestimatethetranche’s“true” risk.Thisisduetothefactthatratingagencymodelsoftenneglect thefattailofcreditportfoliolossdistributions.Inourexamplewecan clearlyseethattheAa2-ratingdoesnotreallyreflectthe“true”riskof theAa2-tranche. Table8.5showstheweightedaveragelife(WAL)ofthefourtranches. For the simulation, we assumed that the CDO matures in 10 years. The WAL for class-A notes is quite low, in part due to the amortization structure of the collateral pool, but to some extent also due to broken Return Range Relative Frequency ©2003 CRC Press LLC coveragetestsleadingtoadeleveragingoftheoutstandingsofthenotes. BecauseofthewaterfallstructureillustratedinFigure8.3,themost senior class has to be repaid before lower clas se s receive repayments. This yields the low WAL for class A. We conclude this section by a brief summary. In the discussion above, our calculations showed that it is very dangerous to rely on “average value” considerations like our “back-of-the-envelope” calculation. Only a full Monte Carlo simulation, based on portfolio models as introduced in this book, will unveil the downside risks and upside chances of an investment in a CDO. 8.3 CDOs from the Modeling Point of View In this section, a general framework for CDO modeling is presented. Not all structures require all elements mentioned in the sequel. In some cases, shortcuts, approximations, or working assumptions (e.g. a fixed 14 , pos sibly stress-tested, LIBOR) can be used for evaluating a CDO quicker than by means of implementing a simulation model where all random elements are also drawn at random, hereby increasing the complexity of the model. In our presentation, we will kee p a somewhat abstract level, because going into modeling details or presenting a fully worked-out case study is beyond the introductory scope of this chapter. However, we want to encourage readers 15 involved in ABS transactions to start modeling their deals by means of a full Monte Carlo simulation instead of just following the common practice to evaluate deals by stress tests and the assumption of fixed loss rates. The example in the previous section demonstrates how dangerous such a “shortcut model” can be. The evaluation of CDO transactions involves three major steps: 14 For example, if in the documentation of a structure one finds that fluctuations of LIBOR are limited by a predefined cap and floor, then one can think of stress testing the impact of LIBOR variations by just looking at the two extreme scenarios. 15 As far as we know, most major banks use, additionally to the “classic” approaches and rating agency models, CDO models based on Monte Carlo simulation comparable to the approach we are going to describe. ©2003 CRC Press LLC FIGURE 8.6 CDO modeling scheme. Collateral Pool / Reference Portfolio Parameters: • DP Distribution • Exposure Distrib. • Industry Distrib. • Country Distrib. • Recoveries • Maturities • Amortization Profiles • etc. Performance: • Returns IRR, ROIC • DP of tranches • EL of tranches • Loss on Principal • Loss on Interest • Excess Spread • Maturity of Tranche • etc. Structured Notes / Tranched Securities Structural Mapping Structure : • Tranching • Cash Flows • Coverage Tests • Triggers • Overcollateralization • Credit Enhancements • Fees • Maturity • etc. „Probability Space“ „Random Variable‘‘ „Distribution“ )),(,( PF t Ω X 1− XP • underlying models: factor model, interest rate model, portfolio model, etc. • market data: macroeconomic indices, interest rates, indices, etc. ©2003 CRC Press LLC 1.Step:Constructingamodelfortheunderlyingportfolio Underlyingthestructureisalwaysanassetpool,forexamplea referenceportfoliooracollateralpool.Thestructuralelementsof theconsidereddealarealwayslinkedtotheperformanceofthe underlyingassetpool,soitisnaturaltostartwithaportfolio modelsimilartothosepresentedinChapters1-4.Additionally, such a mo del should include • multi-year horizons due to maturities longer than one year, • a sound factor model for measuring industry and country diversification in an appropriate manner, and • a model for short term interest rates for capturing the inter- est rate risk of floating rate securities and notes. This first step is the only part involving probability theory. The second and third step are much more elementary. 2. Step: Modeling the cash flows of the structure Based on Step 1, the cash flows of the structure conditioned on the simulated s cenario from the portfolio model representing the performance of the collateral securities should be modeled by taking all cash flow elements of the structure, including • subordination structure, • fees and hedge premiums, • principal and interest waterfalls, • coverage tests (O/C and I/C), • credit enhancements (e.g. overcollateralization), • triggers (e.g. early amortization, call options), etc., into account. From a programming point of view, Step 2 consists of implementing an algorithm for “distributing money” (e.g., in a cash flow CDO the cash income from the collateral securities) into “accounts” (some specified variables reflecting, e.g., principal and interest accounts) defined by the contract or documentation of the deal. Such an algorithm should exactly reflect the cash flow mechanisms specified in the documentation, because leaving out just a single element can already significantly distort the simula- tion results towards wrong impressions regarding the p erformance ©2003 CRC Press LLC ofthestructure.Inadditiontoacashflowmodel,adiscount- ingmethod(e.g.,arisk-neutralvaluationmodelincasethatthe risks,e.g.,thedefaultprobabilities,ofthecollateralsecurities aredeterminedaccordingtoarisk-neutralapproach)shouldbe inplaceinordertocalculatepresentvaluesoffuturecashflows 3.Step:Interpretingtheoutcomeofthesimulationengine Afterthesimulation,theoutcomehastobeevaluatedandinter- preted.Becausetheperformanceofthestructureissubjectto randomfluctuationsbasedontherandomnessofthebehaviour ofthecollateralsecurities,thebasicoutcomeofthesimulation willalwaysconsistofdistributions(e.g.,returndistributions,loss distributions,etc.);seeFigure8.5andthediscussionthere. Figure8.6illustratesandsummarizesthethreestepsbymeansofa modelingscheme. In[37],FingercomparesfourdifferentapproachestoCDOmodeling, namely a discrete multi-step extension of the CreditMetrics TM port- folio model, a diffusion-based extension of CreditMetrics TM , a copula function approach for correlated default times, and a stochastic de- fault intensity approach. The first two mentioned approaches are both multi-step models, which will be briefly discussed in the next section. The basic methodology underlying the third and fourth approach will be outlined in two subsequent sections. 8.3.1 Multi-Step Models Multi-step models are natural extensions of single-perio d portfolio models, like the models we discussed in previous chapters. Essentially, a multi-step model can be thought of as many “intertemporally con- nected” single-period m odels successively simulated. Considering the three major valuation steps discussed in the previous section, one could describe the three steps in a multi-step model context as follows: Step 1 defines a filtered probability space (Ω, (F t ), P), where: • Ω consists of the whole universe of possible scenarios regarding the collateral pool and the interest rate model. More prec isely, every scenario ω ∈ Ω is a vector whose components are defined by the possible outcomes of the portfolio model, including a de- fault/migration indicator realization for every collateral security, a realization of LIBOR, etc. ©2003 CRC Press LLC •(F t ) t=1, ,T isafiltrationofσ-algebrascontainingthemeasurable eventsuptothepaymentperiodt.Anyσ-algebraF t canbeinter- pretedasthecollectionofeventsreflectinginformationsknown uptopaymentperiodt.Forexample,F t containstheeventthat uptotimettheportfoliolossalreadycrossedacertainlimit,etc. Here,Trepresentsthefinalmaturityofthestructure. •TheprobabilitymeasurePassignsprobabilitiestotheeventsin theσ-algebrasF t ,t=1, ,T.Forexample,theprobabilitythat uptotimetmorethan20%ofthecollateralsecuritiesdefaulted isgivenbyP(F),whereF∈F t isthecorrespondingmeasurable event. Step2definesarandomvariable X,becauseassoonasascenarioω∈Ω isfixedbythesimulationengine,thedistributionofcashflowscondi- tionalonωfollowsadeterministicworkflowdefinedbythedocumen- tationofthestructure.Thevariable Xisavectorwhosecomponents containthequantitiesrelevantfortheperformanceofthestructure, e.g.,realizedreturnsfornotesinvestors,theamountofrealizedre- payments,thecouponpaymentsmadetonotesinvestors,etc.The distributionP◦ X −1 ofthe“performancevector” Xthenisthefinal output,whichhastobeanalyzedandinterpretedinStep3.Forexam- ple,therelativefrequencyofscenariosinwhichatleastonepromised dollartoamezzanineinvestorhasnotbeenpaid,constitutesthedefault probabilityofthatmezzaninetranche. Thefiltration(F t ) t=1, ,T definesadynamicinformationflowduring thesimulatedlifetimeofthedeal.Forexample,thesimulationstep fromtimettotimet+1willalwaysbeconditionedonthealready realizedpath(the“history”uptotimet).Thisverymuchreflectsthe approachaninvestorwouldfollowduringthetermofastructure:At timetsheorhewilltakeallavailableinformationuptotimetinto accountformakingananalysisregardingthefutureperformanceofthe structure. 8.3.2CorrelatedDefaultTimeModels Themulti-stepmodelisastraightforwardextensionoftheone-period modelswediscussedinpreviouschapterstoamulti-periodsimulation model.Another“bestpractice”approachistogeneratecorrelatedde- faulttimesofthecollateralsecurities.Wealreadydiscussedthisap- proachinSection7.3.Thecorrelateddefaulttimesapproachcalibrates ©2003 CRC Press LLC defaulttimescompatibletoagivenone-yearhorizonassetvaluemodel bymeansofcreditcurves,assignedtothedefaultprobabilityofthecol- lateralsecurities,andsomecopulafunction,generatingamultivariate dependencystructureforthesingledefaulttimes.Itisnotbychance thatthisapproachalreadyhasbeenusedforthevaluationofdefault baskets:Focussingonlyondefaultsandnotonratingmigrations,the collateralpool(orreferenceportfolio)ofaCDOcanbeinterpretedas asomewhatlargedefaultbasket.Theonlydifferenceisthecashflow modelontopofthebasket. Fromasimulationpointofview,thedefaulttimesapproachinvolves muchlessrandomdrawsthanamulti-stepapproach.Forexample,a multi-stepmodelw.r.t.acollateralpoolconsistingof100bonds,would forquarterlypaymentsover10yearsrequire100×10×4simulated randomdrawsineveryscenario.Thesamesituationbymeansofade- faulttimesapproachwouldonlyrequiretosimulate100randomdraws inascenario,namelyrealizationsof100defaulttimesfor100bonds. Thissafescomputationtime,buthasthedisadvantagethatratingdis- tributions(e.g.,formodelingratingtriggers)cannotbeincorporated inastraightforwardmannerasitisthecaseinmulti-stepmodels. Time-consumingcalculationsinthedefaulttimesapproachcould beexpectedinthepartofthealgorithminvertingthecreditcurve F(t)inordertocalculatedefaulttimesaccordingtotheformulaτ= F −1 (N[r]);seeSection7.3.Fortunately,forCDOmodelstheexact time when a default occurs is not relevant. Instead, the only relevant information is if an instrument defaults between two consecutive pay- ment dates. Therefore, the copula function approach for default times can be easily discretized by calculating thresholds at each payment date t = 1, , T according to α t = N −1 [F (t)] , where F denotes the credit curve for some fixed rating, and N[·] denotes the cumulative standard normal distribution function. Clearly one has α 1 < α 2 < . . . < α T . Setting α 0 = −∞, asset i defaults in period t if and only if α t−1 < r i ≤ α t , where (r 1 , , r m ) ∼ N(0, Γ) denotes the random vector of standardized asset value log-returns with ass et correlation matrix Γ. This reduces ©2003 CRC Press LLC [...]... and synthetic [6] • J.P.Morgan: – A “CDO Handbook”, providing a good overview and introduction to CDOs [69] – The JPM guide to credit derivatives, also introducing synthetic CDOs [68] ©2003 CRC Press LLC • Deutsche Bank: – CDO introduction [27] – Introduction to the use of synthetic CLOs for balance sheet management [25] Regarding the academic literature we do not see as many publications for CDO modeling. .. and D Oakes Analysis of Survival Data Chapman and Hall, 1984 [17] J.C Cox, J.E Ingersoll and S.A Ross A theory of term structure of interest rates Econometrica, 53:385–407, 1985 [18] Credit Suisse Financial Products CreditRisk+ – A Credit Risk Management Framework, 1997 [19] P Crosbie Modeling default risk KMV Corporation, http: //www.kmv.com, 1999 [20] M Crouhy, D Galai, and R Mark A comparative analysis... structure of credit risk spreads Review of Financial Studies, 10: 481–523, 1997 [65] R.A Jarrow and S.M Turnbull Pricing options on financial securities subject to default risk Journal of Finance, 50:53–86, 1995 [66] S Jaschke and U K¨chler Coherent risk measures, valuation u bounds, and (µ, σ)-portfolio optimization Finance and Stochastics, 5(2):181–200, 2000 [67] H Joe Multivariate Models and Dependence... Introduction to Copulas Springer, New York, 1999 [100 ] C Nelson and A Siegel Parsimonious modeling of yield curves Journal of Business, 60:473–489, 1987 [101 ] P Nickell, W Perraudin, and S Varotto Ratings- versus equitybased credit risk modeling: an empirical analysis http://www bankofengland.co.uk/workingpapers/, 1999 Working paper [102 ] J Norris Markov Chains Cambridge Series in Statistical and Probabilistic... Masters Credit derivatives and the management of credit risk The Electronic Journal of Financial Risk, 1(2), 1998 [85] McKinsey & Company, German Office CreditPortfolioView 2.0, June 2001 Technische Dokumentation [86] R Merton On the pricing of corporate debt: The risk structure of interest rates The Journal of Finance, 29:449–470, 1974 [87] T Mikosch Elementary Stochastic Calculus – with Finance in... Moody’s Investors Service Stability of Ratings of CBO/CLO Tranches: What Does It Take to Downgrade a CBO Tranche?, November 1998 [92] Moody’s Investors Service Another Perspective on Risk Transference and Securitization, July 1999 [93] Moody’s Investors Service Moody’s Approach to rating multisector CDOs, September 2000 [94] Moody’s Investors Service Moody’s Approach to Rating Multisector CDOs, September... models Journal of Banking and Finance, 24:119–149, 2000 [52] M B Gordy A risk- factor model foundation for ratings-based bank capital rules Draft, February 2001 [53] G R Grimmet and D Stirzaker Probability and Random Processes Oxford University Press, 2nd edition, 1992 [54] G M Gupton, C C Finger, and M Bhatia CreditMetrics – Technical Document Morgan Guaranty Trust Co., http://www defaultrisk.com/pp_model_20.htm,... functions and copulas Kybernetika, 9:449–460, 1973 [115] J R Sobehart and S C Keenan An introduction to marketbased credit analysis Moody’s Risk Management Services, November 1999 [116] Standard & Poor’s Global CBO/CLO Criteria [117] Standard & Poor’s Global Synthetic Securities Criteria [118] Standard & Poor’s Standard & Poor’s Corporate Ratings Criteria 1998 [119] W Stromquist Roots of transition... McNeil, and M Nyfeler Copulas and credit models RISK, 14 (10) :111–114, 2001 [48] J Frye Collateral damage RISK, 13(4):91–94, 2000 [49] J Frye Depressing recoveries RISK, 13(11) :108 –111, 2000 [50] P Georges, A-G Maly, E Nicolas, G Quibel, and T Roncalli Multivariate survival modelling: a unified approach with copulas http://gro.creditlyonnais.fr, 1996 [51] M B Gordy A comparative anatomy of credit risk models... Finance Springer, 1998 [73] S Kealhofer, S Kwonk, and W Weng Uses and abuses of bond default rates KMV Corporation, 1998 [74] H U Koyluoglu and A Hickman RISK, 56, October 1998 ©2003 CRC Press LLC Reconcilable differences [75] A Kreinin and M Sidelnikova Regularization algorithms for transition matrices Algo Research Quarterly, 4(1/2):25–40, 2001 [76] D Lamberton and B Lapeyre Introduction to Stochastic . ,T.Forexample,theprobabilitythat uptotimetmorethan20%ofthecollateralsecuritiesdefaulted isgivenbyP(F),whereF∈F t isthecorrespondingmeasurable event. Step2definesarandomvariable X,becauseassoonasascenarioω∈Ω isfixedbythesimulationengine,thedistributionofcashflowscondi- tionalonωfollowsadeterministicworkflowdefinedbythedocumen- tationofthestructure.Thevariable Xisavectorwhosecomponents containthequantitiesrelevantfortheperformanceofthestructure, e.g.,realizedreturnsfornotesinvestors,theamountofrealizedre- payments,thecouponpaymentsmadetonotesinvestors,etc.The distributionP◦ X −1 ofthe“performancevector” Xthenisthefinal output,whichhastobeanalyzedandinterpretedinStep3.Forexam- ple,therelativefrequencyofscenariosinwhichatleastonepromised dollartoamezzanineinvestorhasnotbeenpaid,constitutesthedefault probabilityofthatmezzaninetranche. Thefiltration(F t ) t=1,. LLC defaulttimescompatibletoagivenone-yearhorizonassetvaluemodel bymeansofcreditcurves,assignedtothedefaultprobabilityofthecol- lateralsecurities,andsomecopulafunction,generatingamultivariate dependencystructureforthesingledefaulttimes.Itisnotbychance thatthisapproachalreadyhasbeenusedforthevaluationofdefault baskets:Focussingonlyondefaultsandnotonratingmigrations,the collateralpool(orreferenceportfolio)ofaCDOcanbeinterpretedas asomewhatlargedefaultbasket.Theonlydifferenceisthecashflow modelontopofthebasket. Fromasimulationpointofview,thedefaulttimesapproachinvolves muchlessrandomdrawsthanamulti-stepapproach.Forexample,a multi-stepmodelw.r.t.acollateralpoolconsistingof100bonds,would forquarterlypaymentsover10yearsrequire100 10 4simulated randomdrawsineveryscenario.Thesamesituationbymeansofade- faulttimesapproachwouldonlyrequiretosimulate100randomdraws inascenario,namelyrealizationsof100defaulttimesfor100bonds. Thissafescomputationtime,buthasthedisadvantagethatratingdis- tributions(e.g.,formodelingratingtriggers)cannotbeincorporated inastraightforwardmannerasitisthecaseinmulti-stepmodels. Time-consumingcalculationsinthedefaulttimesapproachcould beexpectedinthepartofthealgorithminvertingthecreditcurve F(t)inordertocalculatedefaulttimesaccordingtotheformulaτ= F −1 (N[r]);seeSection7.3.Fortunately,forCDOmodelstheexact time. reallyisaweightedsumofsingleassetrisks,ignoringthepotentialfor diversificationeffectstypicallyinherentinaportfolio. Incontrast,ontheCDOside,itistheportfolioriskwhichendangers theperformanceofthestructure.Recallingourdiscussiononcashflow CDOs,weseethatthetranchingofnotesreallyisatranchingofthe lossdistributionofthecollateralpool,takingallpossiblediversification effectsintoaccount.Butdiversificationdecreasestheriskofaportfolio, sothatthepriceoftheportfolioriskmustbelowerthantheprice obtainedbyjustsummingupexposure-weightedsinglerisks.Thisis reflectedbythespreadsonnotesasgiveninTable8.1:Thespreads paidtonotesinvestorsaremuchlowerthanthespreadsearnedonthe bondsinthecollateralpool.Duetotherisktranchingofnotes,the spreadsonseniornotesisevenlower,duetothecreditenhancement bysubordinationprovidedfromnoteswithlowerseniority. ItisexactlythemismatchbetweenthesingleassetbasedWACofthe portfolioandthemuchlowerweightedaveragecoupononthenotesof theCDO,whichcreatesanarbitragespread.Thismismatchisinone partduetodiversificationeffects,andinanotherpartbasedonstruc- turalelementslikesubordinationorothercreditenhancementmecha- nisms.Callingspecialattentiontothediversificationpoint,onecan saythatCDOsare“correlationproducts”. Anexampleregardingarbitragespreadisgiveninthenextsection inthecontextofCDOinvestments.Conceptually,anyoriginatorofan arbitragecashflowCDOkeepingtheCDO’sfirstlosspieceautomati- callytakesontheroleoftheequityinvestor,earningtheexcessspread ofthestructureinitsownpockets.Therefore,wecanpostponethe arbitragespreadexampletothenextsection. 8.2.2TheInvestor’sPointofView VeryoftenbanksareontheinvestmentsideofaCDO.Inmanycases, ABSbondsofferinterestingandattractiveinvestmentopportunities, butrequire(duetotheircomplexity)carefulanalyticvaluationmethods forcalculatingtherisksandbenefitscomingwithanABSinvestment intothebank’sportfolio.Thiswillbemadeexplicitbymeansofthe followingexample. RecallthesamplecashflowCDOfromTable8.1.Inthisexample we