an introduction to credit risk modeling phần 9 pdf

Note that as T → ∞ the values of µ and η 2 converge to fixed values, and the distribution of x T converges to a steady-state stationary distri- bution. With this framework we can find the price of a European call CSO. Let C(x, r, T ) denote the value of the option. The payoff func- tion for this option is simply H(x) = max(e x −K, 0). The closed-form solution for the call CSO is given by C(x, r, T ) = p(r, T )  e µ+η 2 /2 N(d 1 ) − KN(d 2 )  . Here, N(·) is the cumulative standard normal distribution, p(r, T) is a riskless discount bond, and d 1 = −log(K) + µ + η 2 η , d 2 = d 1 − η. The value of a European put CSO is P (x, r, T) = C(x, r, T ) + p(r, T )  K − e µ+η 2 /2  . The option formula has some similarities with the Black-Scholes op- tion pricing formula. However, the value of a call option can be less than its intrinsic value even when the call is only slightly in the money. This surprising result is due to the mean reversion of the credit spreads. When the spread is above the long-run me an, it is expected to decline over time. This can not happen in the B-S model because the un- derlying asset must appreciate like the riskless rate in the risk-neutral framework. The delta for a call is always positive, as in the B-S frame- work, but the delta of a CSO call decreases to zero as the time until the expiration increases. A change in the current credit spread is heavily outweighed by the effects of mean reversion if the expiration date of the call is far in the future. A credit spread collar combines a credit spread put and a credit spread call. An investor that wishes to insure against rising credit spreads by buying a credit spread call can reduce the cost by selling a credit spread put. In a credit spread forward (CSF), counterparty A pays at time T a pre-agreed fixed payment and receives the credit spread of the reference asset at time T . Conversely, counterparty B receives the fee and pays the credit spread. The fixed payment is chosen at time t < T to set the initial value of the credit spread forward to zero. The credit spread forward can also be structured around the relative credit spread between two different defaultable bonds. Credit ©2003 CRC Press LLC FIGURE7.7 Creditspreadswap. spreadforwardscanbecombinedtoacreditspreadswap(Figure7.7) inwhichonecounterpartypaysperiodicallytherelativecreditspread, (S 1 (t)−S 2 (t)),totheother. 7.5Credit-linkedNotes Credit-linkednotesexistinvariousformsinthecreditderivatives market;see[23,98,68,26].Initsmostcommonform,acredit-linkednote (CLN)isasyntheticbondwithanembeddeddefaultswapasillustrated inFigure7.8. CLNs are initiated in several ways. In the following we outline four examples of typical CLN structures. The first case we present is the situation of an (institutional) investors who wants to have access to a credit exposure (the reference asset) for which by policy, regulation, or other reasons he has no direct access. In such cases, a CLN issued by another institution (the issuer) which has access to this particular credit exposure offers a way to evade the prob- lems hindering the investor to purchase the exposure he is interested in. The issuer sells a note to the investor with underlying exposure equal to the face value of the reference asset. He receives the face value of the reference asset as cash proceeds at the beginning of the transaction and in turn pays interest, including some premium for the default risk, to the investor. In case the reference asset experiences a credit event, the issuer pays to the investor the recovery proceeds of the reference asset. The spread between the face value and the recovery value of the reference asset is the investor’s exposure at risk. In case no credit Reference Asset1 Bank A s1 Bank B Reference Asset2 s2 ©2003 CRC Press LLC FIGURE 7.8 Example of a Credit-linked Note. • Bond • Loan • Portfolio Reference Asset Issuer • Bank • SPV • etc. Seller of Risk Notes Investor • Bank • Insurance • etc. Buyer of Risk Premium Interest Rate / FX Swap Credit Default Swap Contingent Payment Proceeds Interest / Premium Default Risk-Linked Cash Collateral or Purchase of Collateral Securities ©2003 CRC Press LLC eventoccurredduringthelifetimeofthereferencenote,theissuerpays thefullprincipalbacktotheinvestor.Sointhisexampleonecould summarizeaCLNasasyntheticbondwithanembeddeddefaultswap. Inoursecondexample,aninvestor,whohasnoaccesstothecredit derivativesmarketorisnotallowedtodooff-balancesheettransactions, wantstoinvestinacreditdefaultswap,sellingprotectiontotheowner ofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLN inthesamewayasdescribedinourfirstexample.Notethatfrom theinvestor’spointofviewtheCLNdealdiffersfromadefaultswap agreementbythecashpaymentmadeupfront.Inadefaultswap,no principalpaymentsareexchangedatthebeginning. Anothercommonwaytoset-upaCLNisprotectionbuying.Assume thatabankisexposedtothedefaultriskofsomereferenceasset.This couldbethecasebymeansofanassetonthebalancesheetofthebank orbymeansofasituationwherethebankistheprotectionsellerina creditdefaultswap.Inbothcasesthebankhastocarrythereference asset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to some investor who pays the exposure of the reference asset upfront in cash to the bank and receives interest, including some premium reflecting the riskiness of the reference asset, during the lifetime of the note. If the reference asset defaults, the bank suffers a loss for its balance sheet asset (funded case) or has to make a contingent payment for the default swap (unfunded case). The CLN then compensates the bank for the loss, such that the CLN functions as an insurance. In this example, the difference between a CLN and just another default swap arise s from the cash proceeds the bank receives upfront from the CLN investor. As a consequence, the bank is not exposed to the counterparty risk of the protection selling investor. Therefore, the credit quality of the investor is of no relevance 2 . The proceeds from the CLN can be kept as a cash collateral or be invested in high-quality collateral securities, so that losses on the reference asset will be covered with certainty. OurlastexamplereferstoChapter8,whereCLNswillbediscussed as notes issued by a special purpose vehicle (SPV) in order to set-up a synthetic CDO. In this c ase , CLNs are used for the exploitation of regulatory arbitrage opportunities and for synthetic risk transfer. 2 Of course, for a short time at the start of the CLN their could be a settlement risk. ©2003 CRC Press LLC Besides the already mentioned reasons, there are certainly more ad- vantages of CLNs worthwhile to be mentioned. For example, CLNs do not require an ISDA master agreement, but rather can contractually rely on the term sheet of the notes. Another advantage of CLNs is that not only the investor’s credit quality but also his correlation with the reference asset is of no relevance to the CLN, because the money for the protection payment is delivered upfront. This concludes our discussion of credit derivatives. ©2003 CRC Press LLC Chapter8 CollateralizedDebtObligations Collateralizeddebtobligationsconstituteanimportantclassofso- calledassetbackedsecurities(ABS),whicharesecuritiesbackedby apoolofassets.Dependingontheunderlyingassetclass,ABSin- cludevarioussubclasses,forexampleresidentialorcommercialmort- gagebackedsecurities(RMBS,CMBS),tradereceivablesABS,credit cardABS(oftenintheformofso-calledCCMasterTrusts),consumer loanABS,andsoon.Notlongago,itstartedthatABSwerealso structuredbasedonpoolsofderivativeinstruments,likecreditdefault swaps,resultinginanewABSclass,so-calledcollateralizedswapobli- gations(CSO).Ingeneral,onecouldsaythatABScanbebasedon anypoolofassetsgeneratingacashflowsuitableforbeingstructured inordertomeetinvestor’sriskpreferences. Asanimportantdisclaimeratthebeginningofthischapter,wemust saythatadeepertreatmentofABSwouldeasilyjustifythebeginning ofanewbook,duetothemanydifferentstructuresandassetclassesin- volvedintheABSmarket.Moreover,itisalmostimpossibletocapture the“full”rangeofproductsintheABSmarket,becauseinmostcases anewtransactionwillalsocarrynewinnovativecashflowelementsor specialfeatures.Therefore,ABStransactionshavetobeconsideredon acarefulcase-by-caseanalysis. ThischapterpresentsanintroductiontoCDOmodelingina“sto- ryline”style.Somereferencesforfurtherreadingaregiveninthelast sectionofthischapter. 8.1IntroductiontoCollateralizedDebtObligations Figure8.1showsasegmentationoftheCDOmarket.Thereare basically two types of debt on which CDOs are based, namely bonds and loans, constituting ©2003 CRC Press LLC •Collateralizedbondobligations(CBO): Inthiscase,thecollateralpoolcontainscreditriskybonds.Many oftheCBOswecurrentlyfindinthemarketaremotivatedby arbitragespreadopportunities,seeSection8.2.1. •Collateralizedloanobligations(CLO): Herethecollateralpoolconsistsofloans.Regulatorycapital relief,cheaperfunding,and,moregeneral,regulatoryarbitrage combinedwitheconomicrisktransferarethemajorreasonsfor theoriginationofCLOsbybanksallovertheworld,seeSection 8.2.1. Besidesthesetwo,CSOs(seetheintroductoryremarks)areofincreas- ingimportance.Theiradvantageisthereductionoffundingcosts, becauseinsteadoffundedinstrumentslikeloansorbonds,thecash flowsfromcreditderivativesarestructuredinordertogeneratean attractivearbitragespread.AsecondadvantageofCSOsisthefact thatcreditderivativesareactivelytradedinstruments,suchthat,based onthefairmarketspreadofthecollateralinstruments,afairpriceof theissuedsecuritiescanbedetermined,forexample,bymeansofa risk-neutralvaluationapproach. AnotherclassofCDOsgainingmuchattentionaremultisectorCDOs. Inthiscase,thecollateralpoolisamixtureofdifferentABSbonds, high-yieldbondsorloans,CDOpieces,mortgage-backedsecurities, andotherassets.MultisectorCDOsaremoredifficulttoanalyze, mainlyduetocross-collateralizationeffects,essentiallymeaningthat bondsissuedbyadistressedcompanycouldbecontainedinmorethan oneinstrumentinthecollateralpool.Forexample,“fallenangels” (likeEnronnottoolongago)typicallycauseperformancedifficulties simultaneouslytoallCDOscontainingthisparticularrisk.Cross- collateralizationcanonlybetreatedbylookingattheunionofall collateralpoolsofallinstrumentsinthemultisectorpoolsimultane- ouslyinordertogetanideaabouttheaggregatedriskofthecombined portfolios.Then,basedoneveryaggregatedscenarioinaMonteCarlo simulation,thecashflowsofthedifferentinstrumentshavetobecol- lectedandcombinedinordertoinvestigatethestructuredcashflows ofthemultisectorCDO. Thecreditriskmodelingtechniquesexplainedinthisbookcanbe usedformodeling(multisector)CDOs.Ofcourse,asoundfactor model,liketheoneexplainedinSection1.2.3,isanecessaryprerequi- site for modeling CDOs by taking industry and country diversification ©2003 CRC Press LLC FIGURE 8.1 Classification of CDOs. CDOs arbitrage- motivated balance sheet- motivated cash flow structure synthetic structure cash flow structure synthetic structure market value structure Assets high-yield bonds/loans derivatives multisector Assets credit risky instruments ©2003 CRC Press LLC effectsintoaccount.Moreover,inmanycasesoneadditionallyhasto incorporateaninterestratetermstructuremodelinordertocapture interestrateriskincaseoffloatingratenotes. Ingeneral,MarketvalueCDOsaremoredifficulttotreatfroma modelingpointofview.Thesestructuresaremorecomparabletohedge fundsthantotraditionalABSstructures.InamarketvalueCDO,the portfoliomanagerhastherighttofreelytradethecollateral.Asacon- sequence,amarketvalueCDOportfoliotodaysometimeshasveryfew incommonwiththeportfolioofthestructureafewmonthslater.The performanceofmarketvalueCDOscompletelyreliesontheportfolio manager’sexpertisetotradethecollateralinawaymeetingtheprin- cipalandinterestobligationsofthestructure.Therefore,investorswill mainlyfocusonthemanager’sdealtrackrecordandexperiencewhen decidingaboutaninvestmentinthestructure.Thedifficultiesonthe modelingsidearisefromtheunknowntradingstrategyoftheportfolio managerandtheneedofthemanagertoreacttoavolatileeconomic environment.Suchsubjectiveaspectsaredifficultifnotimpossibleto modelandwillnotbetreatedhere. 8.1.1TypicalCashFlowCDOStructure Inthissection,weexplainatypicalcashflowCDOtransaction;see Figure8.2.Forthispurposewefocusonsomeofthemainaspects without going too much into details. • At the beginning there will always be some pool of credit risky assets. Admittedly, it will not always be the case that the pool intended to be securitized was existent at the originating bank’s balance sheet for a long time; instead, there are many cases where banks purchased parts of the pool just a few months before launching the transaction. Such purchases are typically done in a so-called ramp-up period. • In a next step, the asse ts are transferred to an SPV, which is a company set-up especially for the purpose of the transaction. This explains the notion special purpose vehicle. An important condition hereby is the bankruptcy remoteness of the SPV, essen- tially meaning that the SPV’s own bankruptcy risk is minimized and that the SPV will not default on its obligations because of bankruptcy or insolvency of the originator. This is achieved by a ©2003 CRC Press LLC ©2003 CRC Press LLC FIGURE 8.2 Example of a cash flow CDO transaction. ©2003 CRC Press LLC [...]... securitized portfolio are much lower than they used to be Moreover, if an economic risk transfer is achieved, EL costs and EC costs will be reduced to an extent reflecting the amount of risk transferred to the capital market Both effects, and additional tax and other benefits can help a bank to refinance a loan portfolio at much better conditions than it was the case before the securitization 8.2.1.4 Arbitrage... structure and all payments to notes investors senior to the equity piece From the discussion above it follows, that subordination is kind of a structural credit enhancement For example, in a structure with only one equity, mezzanine, and senior tranche, the senior note holders are protected by a cushion consisting of the equity and mezzanine capital, and the mezzanine tranche is protected by the equity tranche... the markets due to inappropriate regulation by the regulatory authorities For example, as indicated in Section 1.3, regulatory capital for bank loans is still not risk- adjusted This results in “pricing distortions”, due to the fact that the capital costs of a loan are independent of the credit quality of the borrower By means of a simple illustrative example we now want to show how regulatory arbitrage... on an average lifetime consideration For measuring economic risk transfer and securitization effects on RAROC more accurately, much more work and modeling efforts are required, very often accompanied by strong assumptions, e.g., regarding the evolution of the reference pool 8.2.1.3 Funding at Better Conditions Funding is an important issue for banks Because every loan needs to be backed by regulatory... a loan to a customer can be too high for making the lending business profitable But if loans are pooled into portfolios for securitization, funding a loan can get significantly cheaper The reasons why a securitization makes funding cheaper, are basically given in the two sections above: Because regulatory capital is reliefed, equity costs of the securitized portfolio are much lower than they used to be... (before and after securitization, respectively), and ∆qα = qα (L) − qα (LSec ) These calculations are sufficient for capturing the securitization impact on the source portfolio Risk transfer” refers to the possibility to reduce the required capital cushion against losses of a portfolio by means of a securitization “Economic” risk transfer happens, if the risk transfer can be measured in terms of the EL and... helping the investors to identify and quantify potential sources of risk and financial distress of the structure This concludes our description of cash flow CDOs for now 8.1.2 Typical Synthetic CLO Structure In contrast to cash flow CDOs, synthetic CDOs do not rely on the cash flows of the collateral pool Instead, credit derivatives, e.g., creditlinked notes, are used to link the performance of securities... equity Note that although we are not proposing to steer a bank 9 Net of liquidity, funding, and administration costs ©2003 CRC Press LLC w.r.t RoE benchmarks, for measuring just the regulatory arbitrage gain of a transaction the RoE gives an acceptable indication To continue the story, let us now assume that the bank decides to securitize the subportfolio by means of a synthetic CLO as illustrated in Figure... bank, whereas all tranches senior to the FLP are placed in the market The regulatory capital after Securitization is then given by RCSec = V × 1.5% × 100% + V × 85% × 20% × 8% = V × 2.86% , because for capital covered by a credit default swap agreement with an OECD bank, regulatory authorities typically allow for a risk weight of 20%, and the funded part of the transaction is eligible for a zero risk. .. originator for the volume of the reference portfolio which is not covered by the senior and junior swaps • In order to guarantee the contingent payments to the originator in case of credit events in the reference pool, the SPV has to invest some money in collateral securities Then, in case of a credit event, the contingent payments from the SPV (protection seller) to the originator (protection buyer) can . LLC eventoccurredduringthelifetimeofthereferencenote,theissuerpays thefullprincipalbacktotheinvestor.Sointhisexampleonecould summarizeaCLNasasyntheticbondwithanembeddeddefaultswap. Inoursecondexample,aninvestor,whohasnoaccesstothecredit derivativesmarketorisnotallowedtodooff-balancesheettransactions, wantstoinvestinacreditdefaultswap,sellingprotectiontotheowner ofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLN inthesamewayasdescribedinourfirstexample.Notethatfrom theinvestor’spointofviewtheCLNdealdiffersfromadefaultswap agreementbythecashpaymentmadeupfront.Inadefaultswap,no principalpaymentsareexchangedatthebeginning. Anothercommonwaytoset-upaCLNisprotectionbuying.Assume thatabankisexposedtothedefaultriskofsomereferenceasset.This couldbethecasebymeansofanassetonthebalancesheetofthebank orbymeansofasituationwherethebankistheprotectionsellerina creditdefaultswap.Inbothcasesthebankhastocarrythereference asset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to. future. A credit spread collar combines a credit spread put and a credit spread call. An investor that wishes to insure against rising credit spreads by buying a credit spread call can reduce. LLC eventoccurredduringthelifetimeofthereferencenote,theissuerpays thefullprincipalbacktotheinvestor.Sointhisexampleonecould summarizeaCLNasasyntheticbondwithanembeddeddefaultswap. Inoursecondexample,aninvestor,whohasnoaccesstothecredit derivativesmarketorisnotallowedtodooff-balancesheettransactions, wantstoinvestinacreditdefaultswap,sellingprotectiontotheowner ofsomereferenceasset.ThiscanbeachievedbyinvestinginaCLN inthesamewayasdescribedinourfirstexample.Notethatfrom theinvestor’spointofviewtheCLNdealdiffersfromadefaultswap agreementbythecashpaymentmadeupfront.Inadefaultswap,no principalpaymentsareexchangedatthebeginning. Anothercommonwaytoset-upaCLNisprotectionbuying.Assume thatabankisexposedtothedefaultriskofsomereferenceasset.This couldbethecasebymeansofanassetonthebalancesheetofthebank orbymeansofasituationwherethebankistheprotectionsellerina creditdefaultswap.Inbothcasesthebankhastocarrythereference asset’sdefaultrisk;seeFigure7.8.ThebankcannowissueaCLN to some investor who pays the exposure of the reference asset upfront in cash to the bank and receives