Credit Default Swaps a Cash Flow Analysis

7/22/2019 Credit Default Swaps a Cash Flow Analysis

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C redit D efau ltS w a p sA C ash F low A na lysisTER R Y B ENZS C HA WEL A ND A LP ER C OR LU

T E R R Y B E N Z S C H A W E Lis amanaging director atCitiInstitutional Clients Groupin New YorkCity NY .terry l benzschawel citi com

A L P E R C O R L UIS an associate at Citi Insti-tutional Clients Group111 N ow York City [email protected]

Ac r e d i t d e f a u l t s w a p ( C D S ) c o n t r a c tis a n a g r e e m e n t t o e x c h a n g e a s p e c -i fi ed s et o f c o u p o n p a y m e n t s i nre turn for t he r igh t t o rece ive thepar face value of a reference obl igat ion af ter thep a r t i c u l a r o b l i g o r u n d e r g o e s a c r e d i t e v e n t .The pa r t i e s i nvolved in the cont rac t a re a pr o-te tionbuyer who pays the coupons orpremiums(usually quarterly), and aprote tionseller whoreceives the p remium s, but must pay the buyerthe par value of an eligible security in exchangefor that security in the event of a default, bank-ruptcy, or restructuring.'In more recent versions ofthe standardCD S contract, the pro tecdon buyer makes anupfront payment set by the seller and pays astandard running 100 bp or 500 bp premiumthat depends on the riskiness ofthe referenceobligor. (Even for the new contract, the upfrontcash flow and fixed spread premium can beconverted to an effective spread premium).The CDS contract is usually obligor-specific,referring to either a corporate or sovereignentity. The securities that are eligible fordelivery to the protection seller in event ofdefault,thereferen e o b l i g a ti o n s retypically froma single class of debt (unsecured bonds, loans,or subordinated bonds, etc.), but can be assetspecific.

For most purposes, both the CDS contractwith upfi-ont payment and standard coupon andthat wdth no upfront payment and market-basedcoupons can be represented using the diagram

in Exhibit 1. The exh ibit shows an examplea CDS written on $10 million of notional wreference to firm XYZ. The buyer of prottion makes quarterly payments, the/^rc/iiii/n/for as long as there is no credit event or unthe maturity of the contract, whichever comfirst. The CDS premium, even if trading wconstant coupon and upfront fee, is oftexpressed as an annual amount in basis poinAlso,contracts from given firm are com moissued at a num ber of standard maturities, wthe most common term being five years. Tprotection seller agrees to pay the buyer tface value of the CDS contract if XYZ undgoes a redit event and the buyer of protedelivers to the seller the defaulted security its cash equivalent.

The advantages of having liquid credefault swap market are well known. Priorthe development of the CD S m arket, investhad few options for hedging existing creexposures or entering a short credit positioIn such cases, investors would have to borrobonds in an over-the-counter market, beisubject to poor liquidity and high financicosts.In addition, typical fixed-rate corporbonds have huge exposure to interest-ramovements, which is often undesirable finvestors wishing to make pure credit plaThe relatively tight bid-ask spreads of CDcontracts and, in particular, CDS index proucts have provided extremely efficient meafor investors to express views on credits

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E X H I B I T 1Representation of a Typical CDS Contract Prior to April 2 9

PremiumLegNotional:US 10MIVI C^'^ ^''Obligor:XYZ

P*Maturity)Term:5Years

Protection Buyer(SellsCDS)

NotionalxSpread Premium(bp)

Reference Obligation( RV) Protection Seller(BuysCDS)Face ValueofObligation ($100)

ContingentLeg(Only ExecutedinResponsetoCredit Event)

otes Tlie protection buyer makes regular coupon payments to the protection seller atid the contingent exchange of a reference obligation in exchange for paymentof its face value in case of a credit event. More recent contracts have an upfront cash payment to the seller of protection and either a 100 bp or. OO bp runningcoupon but can still be represented as in the exhibit.

firms and countries around the world. Furthermore, theCDS contract has provided the building block for other,more complicated partitioning of credit exposure via suchsynthetic products as single-tranche CDOs (S-CDOs),whereby investors can express a view on credit correla-tion with specific risk profiles, and options on CDS,whereby investors take positions on credit spread volatility.Despite the wide success ot CDS contracts as finan-cial instruments, recent turmoil in the credit markets haveexposed vulnerabilities in the CDS market. For example,the lack of a central clearinghouse for CDS trades hasrevealed systemic and firm-specific weaknesses in theability to effectively manage counterparty risk. In addi-tion, cotwentions for trading CDS have enabled tradingpractices that, by enabling investors to go short with littleor no initial investment, have contributed to the unprece-dented volatility in cash and synthetic credit markets sincemid-2007. Furthermore, pressure from buyers of protec-tion via CDS has been blamed for contributing directly,at least in part, to the failure of somefirms.Although CDShave been widely criticized for their role in the currentcredit crisis, an aspect of the CDS market that has beenlargely unrecognized or overlookedisthat current methodsfor pricing and hedging CDS may be inadequate and/orproblematic. For example, we question some assumptionstliat utiderlie the widespread application of risk-neutralpricing theory to CDS; one of which implies that firms

CDS are synthetic bonds with implicit funding at LIBOR.To that end, we examine the CDS asasset swapmodelatid its potential limitations. In addition, we suggest apricing method based on analyses of specified and expectedCDS cash flows that, combined with estimates of phys-ical default probabilities and recovery values, provides analternative perspective for evaluating CDS risk and rela-tive value.

CASH BOND EQUIVALENT OF CDSIt is nearly axiomatic among CDS investors that a

CDS contract can be replicated by long and short posi-tions in cash bonds by the seller and buyer of protection,respectively, who can borrow the reference obligation inthe repo market (Kumar and Mithal |2001 ] and Kakodkaretal. [2006]). That is theno arbitrageargument of the reltionship between credit default swaps and corporate bondsstates that one can replicate the premium leg oftheCDSwith a long position in the reference obligation combinedwith afixed-for-floatingnterest-rate swap and the payoutleg with a short position in the reference bond atid a repoagreement to borrow that security. For example. Exhibit 2shows the cash flows for each leg of the CDS contract inExhibit 1 represented as an asset-swap. Clearly, replicatitigeach leg of the CDS contract itivolves several operationsby both buyer and seller of protection. The no-arbitrage

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E X H I B I T 2No Arbitrage Model for CDS Where the Buyer of Protection Is Long a Bond Financed at LIBOR along with anInterest Rate Swap and the Protection Seller Is Short the Reference Obligation and Has Borrowed the Bond via Repo

BankMakes5-YearLoan

5-YearInterestRateSwap

PremiumLegPaid WfiiieNoCredit Eventupto Maturity)Notionalx Spread PremiumReferenceObligation ( RV)

Fa^eValue of Obiigatio ii ( 100)

5-YrSwapRateContingentLegOnly Executed inResponsetoCreditEvent)

ReferenceObligation

10 MM ofXYZ5-YearParBond10MM5-Year_Eai-Bond

5-Yearnterest

ateSwap

10ofXYZ5-YearParBond

ote Tliis representation assumes that all parties can finance all transactions at LIBOR and that there is no cost to the protection seller to repo the reference asis

model isvddely,if onlyimplicitly,assumed by most marketparticipants. The asset-swap and CDS equivalence isreflected in theZ-spread, *the common measure of adjustingcash bond spreads for comparison with CDS premiums, andby methods for estimating the cashCDS basisasthe spreadto the interest-rate swap curve (Choudhry [2006]). Giventhe general acceptance of the no-arbitrage argumentbetween cash bonds and CDS and because we argue thatthis argument has limitations,weconsider in detail the no-arbitrage model from the perspectives of both buyers andsellers of protection.

The Protection BuyerA depiction of afive-yearCDS contract as an asset-

swap between the buyer and seller of protection appearsin Exhibit2 Consider first the protection buyer who agreesto make regular premium payments to the protection selleras long as there is no credit event or until the maturity ofthe CDS, whichever comes first. To replicate this with acash bond, the buyer of protection must purchase a five-year bond issued by the reference obligor. For this example.

assume that the bond is afixed-rate nstrument purchasat par. To pay for the security, the protection buyer borows the par amount times the notional from a bank, cit bank,at LIBOR. The coupons from the borrowebond are used for two purposes. First, the obligor enteinto a five-year fixed-for-floating rate swap to generathree-month LIBOR to make quarterly interest paymenon the loan for the bond. The remainder of the coupothe amount above the five-year swap rate, is paid out apremium to the protection seller. The fact that the buyof the bond must finance the transaction at a rate assumeto be LIBOR is the reason that the cash versus CDS basis referenced to the bond s Z-spread.

As long as the bond pays coupons to the buyer protection, the buyer can finance the bond and pass thspread premium to the protection seller. If there are ncredit events prior to maturity of the CDS and the reerence bond, the interest-rate swap expires and thbond s obligor pays the face value to the protectiobuyer, which is used to repay bank 1for the initial loaHowever, if the reference obligor triggers acredit eprior to CDS maturity, the buyer of protection is du

RE I T DEFAULT SWAPS:^ C SH PLOW ANALYSts W I N T E R 2


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the face value of the bond times the notional from theprotectionseller.Th e protection b uyer can use the payoffto deposit in bank 2 earning LIBOR and use theLI BO R proceeds to enter into a floating for fixed-rateswap until the remaining maturity of the initial five-year swap. Th isway,at the maturity of both interest-rateswap contracts, the n et payout will be zero and the bu yerof protection withdraws the deposit in bank 2 and usesthe proceeds to repay the principal on the original loanfrom bank 1.Finally,it should be n oted that the demonstration ofthe no-arbitrage model of the cash bond versus CDS rela-tion is not unique. One could devise other combinationsof bond purchases and sales, borrowing arrangements,interest-rate swap agreements, and repurchase agreementsto equate cash bonds andCDS.The importance of the pre-

sent dem onstration is that all of these mechanisms involvetransactions in markets whose price determinants maydiffer from those of the deliverable obligations, giving riseto non-credit-related influences on both bond spreadsand C DS premium s. Furth ermore, these factors may affectone side of the buyer/seller relationship and not the other.The Protection SellerConsider now the C DS as asset-swap from the per-spective of the seller of protection as depicted in the righ t-

hand side of Exhibit 2. Recall that the protection sellerreceives quarterly payments from the protection buyerunless there is a credit event before the maturity of theCDS.If a credit event is triggered prior to maturity, how-ever, the protection seller must pay the protection buyerthe face value of the reference obligation times th e notionalof the CDS contract. To replicate the payout profile ofthe protection seller in the cash bond market, one can sellshort the reference security, deposit the sale proceeds ina bank at LIB OR , and enter into afixed for floatingswap.Th e fixed leg of the swap is comb ined with the p remiumfrom the protection buyer to pay the coupon on the bor-rowed security. As for the protection buyer, the modelassumes that the seller of protection can borrow and lendat LIBOR and that none ofthesecurities trade as speci lin the repo market.

Assume now that the reference obligor triggers creditevent prior tomaturity.The protection seller receives the ref-erence security in exchange for the face value of the secu-rity times the notional. The seller makes that payment fromfunds deposited in the bank at inception. The reference

security is then delivered to the counterparty in the orig-inal short sale by the protection seller and the borrowedbond is returned sthe repo is unwound whereby the sellerof protection pays the lender the loss on the borrowed bond(i.e.,face value minus recovery).Finally,the fixed-for-floadngrate swap is offset by a floating-for-fixed swap for th eremaining time to maturity.

The CDS Cash Bond BasisThe previous example is used to explain why, in anarbitrage-free setting, the break-even C DS prem iumshould be identical to the asset-swap spread on a bondpriced atpar.Because of this, when investors wish to com -pare market risk premiums between CDS and their ref-erence bon ds, they often use the Z-sprea d, which is a

spread to the LI BOR curve.* Despite their assumed th e-oretical equivalence, Z-spreads on cash bonds and theircorrespo nding default swap spreads are rarely the same, andthe difference between themiscalledthebasis For exampleExhibit 3 shows the CDS minus cash bond basis for firmsin the North American investment-grade CDS index(CDX.NA.IG)' from December 2005 through April2009. From the inception of the C DX index in2003,thebasis had typically been 10 bps-20 bps positive (CDXpremium greater than the average of its constituent's bon dZ-spreads),but since2006,the basis has largely been neg -ative, with average bond spreads exceeding CD X .N A .IGpremiums by as much as 250 bps.

Given the complexity of the relationship betweena firm's reference bond and its CDS, as demonstrated inExhibit 3,it is not surprising that the basis between cashbonds and CDS is rarely zero. In fact, there are a varietyof market factors, technical details, and implementationfrictions that underlie th e basis. Some of the w ell-knownfactors that influence the basis are as follows: M eth od of calculating the basis (e.g., Z-spread ,

I-spread, C-spread) Imbalances betwe en market demand for buyin g andselling protection Differences in hquidity prem iums for a firm's cashand synthetic assets Impact of cheapest to deliver cash asset Funding versus LIB OR Cash reference assets trad ing away from par value Difference in conventio ns for accrued interest onbonds and CDS premiums

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E X H I B I T 3Historical CDS vs. Cash Bond Basis for Firms in the North American Investment-Grade CDS Index,December 2005-May 2009

4 0 0 )D e c - 0 5 J u n - 0 6 D e c - 0 6 J u n - 0 7 D e c - 0 7 J u n - 08 D e c - 0 8 J u n - 0 9 D e c - 0 9

Source:Citi and Markit Partners

Counterparty risk exposure Risk from different definitions of default for cash

and synthetic assets Choice of calculation conventions for the basis and

hedge ratios Bonds trading tight to LIBOR (such as AAA rated

bonds) have non-negative CDS premiumsDetails of the factors underlying the cash versus CDS

basis are described in detail elsewhere (Kumar and Mithal[2001] Choudhry [2006], Kakodkar et al. [2006], Kingand Sandigursky [2007] and Elizalde et al. [2009]), so arenot discussed further in this article. We list these factorsonly to illustrate the large number of factors that influ-ence the CDScash basis. The contributions to the basisfrom the various sources that include funding rates swapspreads, and the repo market provide limitations to theutihty of the no-arbitrage argument asapplied to creditdefault swaps and their cash bond asset-swaps. That is,unambiguously distinguishing the effects of all the fac-tors contributing to the current basis is extremely diffi-cult. Furthermore, hedging all the factors identified asunderlying the basis, even if known, isalso difficult giventhe available market instruments. Finally, as we describein detail in the following sections, there are other seriousobjections to the no-arbitrage explanation for the spreadrelationship between CDS and their cash asset referents.These are discussed after the introduction of the standardmethod for valuing CDS in the risk-neutral setting.

THE STANDARD MODELFOR CDS VALUATION

The standard framework for interpreting CDS valand risk management is thereduced formmodel. Intice the reduced-form model takes as input the marCDS premium and U.S. Treasury rates and solves, irisk-neutral setting, for the default probabihty that resuin equal expected present values of premium and cotingent legs. A popular version of that model has beproposed by Hull and White [2000] based on the reducform approach of Duffie and Singleton [1999| anddetailed description appears in O'Kane and Turnb[2003].In the reduced-form approach the credit evprocess is modeled in continuous time as a hazard rthat represents the instantaneous probability of the fidefaulting at a particular time. Within the risk-neutsetting, the present value of a security is the expecvalue of its cashflowsdiscounted at the risk-free rate. example,l e t the time of default be denoted as Tand assuthatRV is a random amount recovered if default occbefore the end of the period, T(i.e.,T r ) e ' RV

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Within this framework,it hasprovedcon-venient tomodel default events usingaPoissoncounting process asintroduced byJarrowandTurnbull 11995| wherethecumulative risk-neu-tral probability of defaultin theinterval from0 totisgivenby

E X H I B I T 4Representationof theContingent Legofa Credit Default SwapintheRisk Neutral Setting

ex p ju 2where Xis the instantaneous jump to default ordefault intensity Forany given interval from i1 tof, theprobability of defaultbytime/conditionalonsurvivalup totime- 1canbeobtainedasfollows:

3

Because thedefault probabilities inEqua-tions 2)- 3) arederived in the risk-neutral set- -dng, their values can be inferredfixjmmarket pricesof CDS contracts.Asnoted before,aCDS contracthastwocash flow streamsa premiumleg and acontingentleg.The premium leg consists of quarterly fixedpayments madeby theprotection buyerto theseller untilmaturityoruntilacredit event occurs, whicheveris first.Onthecontingent leg, the protecdon seller makes a singlepayment dependenton theoccurrenceofa credit event,usually default. Although thecontingent payment is theface value of the reference bond, the protection buyer mustdeliverthereference securityorits equivalent valueto theprotecdonseller.Thus, the amount of contingent paymentcanbemodeled astheface amount multiplied by 1Rl^,where RVistherecovery rate immediate after default,expressed as percentageofthe face.

The patternofpotential contingent legpayoutsofaCDS isrepresented inExhibit4. Oneach time stepinthe exhibit, a credit event occurs with probabilityp ^^j orsurvives with probability 1 - p _ Asdescribed previ-ously, if the firm defaults, the seller paysanamount whosevalueis theequivalentof the face minus the recoveryvalue, 1RVin Exhibit4, and thecontract terminates.Otherwise, the firm survives untilthenext periodinwhichit again either defaultsorsurvives.Theexhibit illustrateshow marginal defaultandsurvival probabilities accumu-late over timeupundl defaultormaturity.

1- R V-+2 4 T - 1

TimeinQuarters4T

In the risk-neutral pricing framework, the CDSspreadat theinitiation of the contract is assumedtoreflectequal present valuesofthe premiumand thecontingentlegs.Because the protection buyer makes thequarterlypaymentsofamount c/4 wherecis theannualizedpre-miumor CDSspread)conditionalon thesurvivalofthereference entity with probability 1- CPD^,thepresentvalueofthe premiumlegequals

PVp

4

whered^denotestherisk-free discount factor from = 0to tquartersandTisthematurityoftheCDS inyears.Onthecontingent leg, the protection seller makesa pay-ment 1 - RFonly if a credit eventhasoccurredin a par-ticular time interval with probability CPD^CPD .Hence thepresent valueofthe contingentleg iscalcu-latedasfollows:

PV conngent 5

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E X H I B I T 5Cumulative Risk-Neutral Probabilities of Default by Maturity for Various Credit Rating Categories March 19 2

100T

au

0.5 2 3 5Maturity years)

AAA BB - - - B

CCC CSource:Citi and Markit Partners

*I Ji


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failure to account for the cost of risk on the mark-to-market of positions in the replicating portfolioarising from changes in swap and repo rates as wellas changes in either party's credit risk;

differences in the price of protection for holders ofdifferent bonds issued by the same obligor but pricedin the market at different relationships to par;

consistent positive CDS premiums for AAA ratedcredits, which almost always trade at premiums toLIBOR, are inconsistent with the asset-swap analogyofCDS.Profitability of Asset Swapsin a CDS Replicating PortfolioConsider first the mark-to-market sensitivities of a

buyer and seller of protection that enter into each side ofan asset-swap replication of a CDS contract as depictedin Exhibit2.For such investors, the structure of the repli-cating trade will allow them to meet their contractualpayoffs as if each were involved in a CDS trade. However,investors in the replicating long- and short-CDS portfo-lios will have additional exposures, even assuming that thereference bond is trading at par and each side is able toborrow at LIBOR. For example, the buyer and seller ofprotection will be exposed to movements in swap ratesfrom the swap contract. For example, if swap ratesrise,allelse equal, the buyer of protection will have unrealizedprofit on the swap position that it has obtained at a pre-viously lower rate and the seller will have a mark-to-market loss. This is not an immediate issue for eitherinvestor, aside from potential credit exposure and/ormargin calls,because the protection buyer must continueto fund the LIBOR swap from the fixed bond proceedsand the seller of protection will continue to receive theincreased LIBOR from their bank. Another problemresults from the fact that changes in the CDS contractmay not mirror those of the underlying bonds. That is,the investor in an asset-swap is exposed to the CDSversus cashbasis.As shown in Exhibit 3, changes in thebasis can be large and have moved as much as severalhundred basis points over several months.

Consider also what happens if the bond issuer inan asset-swap defaults. The seller of protection mustdeliver the notional times the par face value of the ref-erence obligation to the protection buyer in return forthe reference security. The protection seller passes thereference security to the counterparty to which it has

sold the bond short and terminates the repo (or arrangesa reverserepo .**Assume that, prior to default, swap rateshave increased. Presumably, the price of the fixed-rateborrowed bond will have decreased from par value dueto an overall rise in rates that is independent of its creditquality. This has a couple of effects. First, the borrower ofthe bond is now paying a higher rate than LIBOR on theface value of the bond at its current market price. Thatis,a new buyer would have to make a lower absolute pay-ment to fund that same bond at that lower price. Fur-thermore,aninvestor wishing to enter intoaCDS contracton that bond will have to buy protection on less than fullface value of the bond. Under those circumstances, onemight expect the CDS price to change as marginal buyersof protection for bonds bought at less than par, whosecoupon spread to LIBOR remains that of the originalbuyer, will require less total protection since the CDScontract is written on par face value, as demonstrated inthe following section.

Failure of the Law of One CDS PriceAnother issue with the CDS as asset-swap analogy

is that the value of credit protection on a single obligorwill differ for investors of its bonds with different couponsat the same maturity. For example, consider three bondsx the same issuer whose indicative information appearsin Exhibit6.Assume that Bond 1 is afive-yearbond issuedtoday at par with a coupon of 7 . approximately equalto the yield of a BBH rated bond on June 15,2009. Nowconsider another bond. Bond 2, originally issued with a15-year maturity exactly 10 years ago by the same issueras Bond 1. At the time Bond 2 was issued, the obligorwas a much better credit and was able to issue the bondat par with a coupon of 3.4 . However, now that the

E X H I B I T 6Indicative DataforThree Five-Year Senior U nsecuredBondsatPar Discount andPremium Prices Relativeto Par

BondBond2Bond3

Maturity(Years)555

Price100

85115

Coupon7.03.4

10.6

Yieldr7 o

7.07.0

Recoveryin Default404040

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obligor is BBB rated, investors are demanding a7%yieldand Bond 2 is trading at a discounted price of85.Finally,consider a third bond issued five years ago at par, whenborrowing by B rated credits requiredacoupon of 10.6%.At the current coupon rate of 7%, that bondisnow tradingat 115.

The risk-neutral cumulative probabihty of default,C P D at any maturity for this obligor can be calculatedfrom the obligor's par yield curve and the U.S. Treasuryyield curve. For example, to find the risk-neutral defaultprobability for the 0.5-year par bond,CPD^^.^,we assume arecovery value (in this case 40% of principal) and solve fortheCPD^^that equates the cashflowsfromhe 0.5-year parbond to the value of 100 when discounted by the corre-sponding Treasury yield. This process is repeated at regularintervals over the life of the bond inquestion.The resultingCPD ;^curve for theBBBrated parbondson June 15,2009,appears in Exhibit 7.

Within the CDSasasset-swap model,wecan use thevalues forCPD^at various maturities in Exhibit7to cal-culate the expected premium of a credit default swap viaEquation 6).The value for the premium obtained usingU.S.Treasury discount curves is 414 bp per annum. Wecan approximate this premiumasa yield spread to LIBORby subtracting the difference between LIBOR and Trea-sury spot rates on June 15, 2009, a difference of48bps.Thus,our estimate ofafive-year CDS premium for thisBBB obligor is roughly 366 bps.

Now consider Bond2,the discount bond issued bythe same obligor as Bond 1. As mentioned. Bond 2 was

E X H I B I T 7Risk-Neutral Default Probabilities as a Functionof Maturity Implied by BBB Bonds and U.S. TreasuryPar Yield Curves June 15 2009

2 3Maturity years)

issued 10 years ago as a 15-year bond at 3.4% and nohas 5 years to maturity. Assume that we bought Bondon June 15 ,2009, at 85, a price well below the par vaof Bond 1. Since both Bond 1and Bond 2,while tradat 100 and 85, respectively, will have a 40% recovery face value in default, we might require less default ptection ifweowned Bond 2 than ifweowned BondSince CDS are quoted in units of100points of face valthe need for less protection on Bond 2 should translainto fewer CDS contracts foragiven notional amountbonds than for Bond 1(or an equal number of contraat a lower spread premium).

We determine the necessary premium on a CDS default protection on 100 face of Bond 2. Let N, be tnotional amount of protection that we need to buy to netralize the default risk of Bond2.Tobreak even in defathe buyer of Bond2will need to get45points per 100 fi.e.,85 40) from the protection seller. In terms of Ccontract, this should be equal to N, * (1 - RV/).Thequating these two values, we have

85-40 = N , * 1- RV100

and therefore N^ - 15.Because Bonds 1 and 2 are fthe same obligor, we can solve for the premium,c,usthe same risk-neutral default probabilities and recoverate that we used for Bond 1. The resulting value off275 bps per annum.

In fact, a simple way to determine the CDS prmium adjustment required for a bond trading away fropar is to realize that the relative premium between a pand non-par bond from the same firm is directly prportional to their losses in default. That is,for the examofBonds 1and 2,

8 5 - RK1 0 0 -

45= X f = 275 bp

p.

CDS contracts are quoted in units of100face regaless of the prices of bonds eligible for delivery in defauso what is the correct fair price ofaCDS protection an obligor having issued Bonds1 and2? Clearly,anabased on the risk-neutral credit curve would suggest twdifferent prices for CDS protection per 100 notional Bonds 1 and 2; 366 bps per annum on Bond 1 a

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275 bps per annum on Bond2 .Similarly, for Bond 3 fi-omthis same obligor, having a 10.6% coupon and trading at115 for 100 notional, our analysis suggests that the priceof CDS protection necessary to recover the price in defaultis 458 bps.

One implication of the foregoing analysis is that, fora firm having bonds with different market prices relativeto par, the cost of protecting one's original principal canvary significantly, thereby providing potential arbitrageopportunities. For example, consider three investors. A,B,and C, where A owns 100 face of Bond 1 in Exhibit 6,B owns 100 face of Bond 2, and C owns 100 face ofBond 3. Investor A will pay 100 for Bond 1, whereasinvestor B pays only 85 for Bond 2 and investor C pays 115 for Bond 3, assuming that the bonds all trade atequivalent cash flow yields.'' Suppose each investor buysprotection on their respective investments to cover poten-tial losses from default and, based on the risk-neutraldefault curve in Exhibit 7, assume that the cost of CDSprotection on 100 face is 3.66% per annum. Now, con-sider expected returns from two different hedging sce-narios involving investors A,B,and C.

CaseI:Investors A,B,and C each buy CDS on 100units of face and pay 3.66 a year until maturity ordefault.Case :Investor A again purchases protection of 100units of face at$3.66.However, investorBnow buysCDS protection on only 75% of the outstandingface of the bond purchased at$85.' In default, B wiU

receive 75 from the CDS contract and the recoveryvalue of 40 on the excess 25 points of face notdelivered into the contract(i.e., $10),thereby recov-ering their investment. Investor C, having paid 115for 100 face of bond, buys 125 units of CDS pro-tection to cover potential loss of investment fromdefault. That is, in defluilt, investor C will receive 125 from the CDS contract but must deliver anadditional 25 units of face at a recovery value of 40into the CDS contract at a cost of$10.In each scenario, we assume a constant default rate

of 0.28%(i.e.,the annual historical rate foraBBB credit)over the five-year contract term. Exhibit 8 displays theinternal rates of return (IRRs) for investors A,B,and Cunder two different default scenarios for each of the twohedging scenarios. The first row in Exhibit 8 shows thatinvestor A has an expected IRR of3.4%under averagedefault conditions but a return of only 1.7% if defaultoccurs within the first six months. Consider for com-parison the returns for investorB.In Case 1, buying pro-tection on 100 face ofCDS,IRRs under historical defaultrates for investors A and B are similar. The 0.3%i advan-tage for investor A in C'ase 1 results from the largercoupon on the par bond and occurs despite the 15-pointadvantage in cases of default for investor B. However,this advantage for investorAwill only result on average.That is, as shown in the far right column of Exhibit 8,if the issuer defaults in the first six months, investor B willhave windfall gains of 39%,owing to the 15 points of

E X H I B I T 8Returns from Investing in Hypothetical BBB Rated Par and Non Par Bonds for CDS ProtectionBought for Full Face Value of Bonds or Adjusted for Loss Given Default

InvestorAB(Case1)B (Case2)C (Case1)C (Case2)

BondTypeParDiscountDiscountPremiumPremium

FaceofCDS10010075

100125

CDS Spreadper UnitBond Face

366366275366458

IRR(Histor ica lDefaultRates)

3.43.14 .13.72.9

IRR(DefaultintheFirst6 Months)

1.738 8

0.4- 22 .0

2.7Note:Theaverage cumulative five year default ratefor a BBB rated creditisroughly 1.4 .

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excess return on the 100 face of CDS protection.Exhibit 8 displays IRRs for investor C in Case 1 thatare the reverse of those for investorB .Thatis,for averagedefault scenarios, investor C performs slighdy better thaninvestor A (andB),but for early default, investor C suf-fers a huge loss of22 on the initial $115 investment.

In Case2,investor B buys CDS protecdon on only75 of the outstanding face. Investor B's annual premiumpayments therefore will be only $2.75 per unit face perannum per unit of bond face owned. In this case, theexpected IRR for investor B increases to4.1 ,exceedinginvestor A's IRR of 3.4%. The advantage in expectedreturns to investor B under historical default rates resultsfrom a combination of relatively low default rates and the"puU-to-par" at maturity under conditions of no default.However, should a default occur in the first six months,B's IRR will be only 0.4%, because B, while having hisinvestment protected, receives little benefit from the pullto par at maturity. The expected IRR for investor C inCase 2, owner of the premium bond, is lowest at 2.9%,owing to the large premium required for 125 points ofCDS protection. However, for an early default, investorC has a greater IRR than investor A and investor B in Case2.

The foregoing analysis has several implications forinvestors in bonds and CDS. First, it is clear that therisk-reward aspects of CDS protection will differ forinvestors in par and non-par securidesfixjmhe same issuer.In fact, it is not clear which hedging strategy is best for non-par securities as each has its advantages and disadvantages.Buying and hedging discount bonds appears to be the bestoverall strategy, whereas investing in premium bonds has

the greatest downside risk. More importantly, howevthese examples reveal clear limitations of the CDS as asswap analogy as a general framework for relating boprices to CDS spreads.ANALYSIS OF CDS CASH FLOWS

Civen the limitations of the CDS as asset-swframework described in the previous secdon, we introdua simplified approach to evaluating CDS contracts aexplore its usefulness for interpreting market data on Cspreads. In the proposed cashflow framework, we vthe premium and condngent legs as sums of the discounvalues of their expected cash fiows under physical msure.For example. Exhibit 9 displays expected cash flofrom premium and contingent legs of a hypothetical onyear CDS contract. The valuesof cpdare expected culative probabilities of actual defaults from time f = 0t ." Notice that the size of the premium paymshown on the left-hand side in Exhibit 9 decreases otime, reflecting the fact that their probabilities of paymdecrease as the likelihood of survival decreases over timThe contingent payments appear on the right-hand sof Exhibit 9. Over any quarterly interval, expected averCDS payouts in default are typically smaller than thassociated premiums. That is because the larger payoare weighted by the probabilities of default. In orderdetermine the size of the payouts in default one massume a recovery value for the reference security. In adtion, when valuing the CDS under physical measure must also assume a term structure of physical default ratFor now, we can assume that we know precisely the te

E X H I B I T 9Example of Quarterly Cash Flows from Premium Leg left-hand panel) and Contingent Leg right-hand panel

cpdj0 0.25 0.5 0.75 1 0 0.25 0.5 0.75

(years) (years)Aofe: L is the Loss Given Defaultof aHypothetical One-YearCDS ontract cashflows notdrawntoscale .

50 CREDI T DEFAULT SWAPS:A CASH FLOW ANALYSIS W I N T E R


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structure of physical default rates foralltenors forallcreditrating categories. We will consider the implications ofestimating those probabilities in a later section.To can calculate the present values of the premiumand contingent legs under physical measure using similarequationsasthose for risk-neutral pricin g in Equations (4)and (5), except we substitutecpd^ for CPD such that

4i\PVp 9)

andPV

conungciit {cpd -cpd ,)*{\-RV) (10)

Unlike in the risk-neutral setting there is, a priori,no expected premium that relates the PVs for the con-tingent and premium legs. Our approach is to first calcu-late abreakeven premium, h, as the annualized quarterlypremium necessary to equate the present values of thecontingent and premium legsascalculated using expectedphysical cashflows.That is.

(11)(Note that within the asset-swap framework,h jwould correspond to the CDS premium.)To calculate breakeven CDS premiums, we needestimates of physical default probabilities such as thoseshown in the left-hand panels of Exhibit 10 for severalpoints in the credit cycle. Each plot shows cumulativedefault curves by rating. Default rates are obtained fromhistorical data and combined with values obtained froma market-based Merton-type model as described in theAppendix.'- The physical CPD values at the left inExhibit 10 are used along with the assumption of anaverage 40% recovery value in default to generate thebreakeven CDS premium curves in the correspondingright panels of the ex hibit. Th e top plots are for the highliquidity, tight spread environm ent of March 2006, withvalues in the middle panels taken at cyclical w ide spreads,and those in the lower graphs from a more average spreadand default environment. Notice that default curves are

ordered by ratings and show only slight adjustments withchanges in the credit cycle. However, breakeven spreadpremiums required by sellers of protection vary by morethan a factor of 10 over the cycle.We can compare market CDS spreads with theinferred breakeven spreads to determine CDS risk pre-miums above the calculated compensation for default. That

is, we use market values of CDS premiums to infer theexcess compensation required by sellers of protection, w hoare lon g credit exposure, for their large promised pay-outs in the event of default. Exhibit displays CDS spreadsby tenor and rating category for the same three points inthe credit cycleasin Exhibit 10.Noticethat likebreakevenCD S spreads, market C DS premiums also vary greatly overthe credit cycle. For example, CD S spreads in th e top left-hand panel, obtained during the high-liquidity, pre-crisisperiod, are relatively tight with even CCC spreads below1,000 bps. In contrast, the CDS spreads in the middle panelnear the height of the crisis are nearly a factor of 10 largerthan those three years earlier. Finally, as the crisis abatedin 2010, spreads have returned to roughly their pre-crisislevels,as shown in the bottom panel.

The tables in the right-hand portion of Exhibit 11display by rating and tenor the CDSrisk premiums (i.e.the excess CDS spreads over breakeven values) at the cor-responding points in the creditcycle.Notice that, althoughmost values for the risk premium in the tables are posi-tive,durin g p eriods of relatively low spread levels, the riskpremium s for high-quality names have been negative. O fcourse, it is possible that the relatively small negative riskpremiums reflect errors in our assumed default probabil-ities and/or recovery values. However, some of these neg-ative values are a sizable fraction of their estimatedbreakeven spread prem iums. If correct, these negative riskpremiums suggest that, for relatively high-quality creditsat short tenors, sellers of protection durin g periods of highliquidity did not receive sufficient compensation to covertheir average expected payouts in default. More gener-ally, spread premium s above those required to c om pe n-sate for default can be quite large relative to breakevenspreads, particularly during periods of market stress asshown in the middle table of Exhibit 11.

Itisimportant to note that the inferred negative riskpremiums in the top and bottom tables of Exhibit 11would likely not be evident within the CDSasbon d asset-swap framework. That is,since CDS spreads in the asset-swap framework are thought to be spreads over LIBORrather than simple cashflows adding LIBOR to the CDS

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E X H I B I T 1 0Estimates of Physical Cumulative Default Probabilities left-hand side) and Breakeven CDS Premiumsright-hand side) by Credit Rating and Tenor for Three Different Points in the Credit Cycle

0.01

s 10000

S 1000 : 1

S 10000

1 - >.

lA osa

10000anVara0cOIraSm2 3 4 5 6 7 8 9 10

aturity years)AAA AA A BBB BB

1-Mar-06

1

1 -11

1-Mar-10

aturity years)

B CCC CNote: reakevenCDS premiumsare calculatedusing Equation (6) and thecorresponding default probabilitiesand arecovery ratein default of 40 .Source:Citi and Moody'sInvestors Service

premium would result in a greater spread than those shownfor breakeven. However, it would seem difficult to escapethe conclusion that in these times of high liquidity andtight credit spreads, sellers ofCDSprotection have beenundercompensated for their risk.

SUMMARY AND IMPLICATIONSWe examined the assumption that credit default

swap (CDS) contracts can be replicated by bonds fundedat LIBOR along with an interest-rate swap and a repur-chase agreement. We find limitations of the bond-CDS

equivalent hypothesis including difficulties in replicatia bond synthetically in the CDS and repo markets adifferent implicit CDS premiums for an obligor s bonof the same maturity but trading at different prices retive to par. We also examined an alternative approachCDS valuation, one that uses assumed physical defarates as opposed to risk-neutral default rates. The implicash flows from each leg under physical measure wethen discounted at appropriate risk-free rates and tresulting breakeven premiums were compared to tmarket-implied CDS spreads. Results were presented various rating categories and tenors for several points

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E X H I B I T 1 1CDS Spreads and Difference between CDS Spreads and reakeven Spreads by Rating Categoryand Tenor for Three Different Points in the Recent Credit Cycle

10000

10Maturity years)AAA - - - AA - A BBB- - - BB - - B CCC/C

Market PremiumsMinusMaturityAAAAAABBBBBBCCC C

MaturityAAAAAABBBBBBCCC C

MaturityAAAAAABBBBBBCCC C

March11

- 105

3518

546March1112

109111194457

12445665

Breakeven1 2006

33

-11-20-18

1835

6092 2009

3757167

136389

57

- 8-11

- 46198

707

5757177

153381

1191 10944897

March1

1715152675

241619

45441 2010

3- 1

-28-24-10

81362767

57

-10- 712

141436891

CDS Spreads7

12- 2

11087

125705

7837590

162366

1 7544 9

71839

30156442847

10175

1425

108155700

108480

1011 169

373994

3478

1032202648

168443788

Source:Citi and Markit Partners Inc

the credit cycle. The methodology enables measurementof the minimum amount of risk premium compensationthat must be required by the protection seller to breakeven for expected default. Our analyses reveal that, attimes, market CDS premiums for high-quality creditswere often insufficient to compensate sellers of protectionfor expected payouts from default. Furthermore, analysesof CDS cashflowsunder physical measure highlight asym-metries between risk premiums received by investors inbonds versus CDS that are not evident from similar com-parisons within the CDS as asset-swap framework.

A P P E N D I XESTIM TING PHYSIC L DEFAULTPROBABILITIES

Acritical assumption underlying the proposed method-ologyis that cumulative default functions over time are knownfor all credits. Fortunately, the credit rating agencies, such asMoody sand Standard Poor's,have compiled extensive sta-tisticson cumulative default rates for issues with given initialratings.A summary of those ratings appears in Exhibit Al.Cumulativerates by year are shown out to 15years,but dataare availablefor out to 30years. Thus,byknowing the agency

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E X H I B I T A lCu mu lative De fault P roba bilities for Bonds by Ra ting Category as a Fun ction of Years since Issuan ce, 1920-20

AA AAAABBBBBBCC C

1

1 21 25 5

29 7

2

2 83 5

12 639 1

3

1 31 36 5

19 145 3

4 1 2 52 19 4

24 549 6

5 ? 3 72 9

1228 654 9

6 4 413 7

14 73257 5

7

61 24 3

173558 5

8OR 81 55

19 237 859 2

9 7 91 75 6

21 339 963 2

17

126 3

22 942 266 5

11 71 22 27

24 444 266 5

12 71 32 37 7

25 645 866 5

13 71 42 58 4

26 747 666 5

14 71 52 69 3

27 549 567 9

15 71 62 9

1 228 55 967 9

Source: Dataarefrom Moody s nvestors Service.

credit rating (or an analyst's or mode l equivalentrating),one canderive the probabilities of default for each successive time period.One problem with using historical default rates for deter-mining default probab ilities is that default rates are credit-cyc le-dependent, ranging from over12%in som e years to less than 1%in others (Altman [2003]). Thu s, in practice, we use a M erto n-based contingent claims analysis model to derive estimates ofdefault probabilities from one to five years and historical valuesafter that. For example. Exhibit Al shows default probabilitiesout to 5years constructedas acombination of marginal defaultrates in years 15 from Sobehart and Keenan's Hybrid Proba -bility of Default (HPD) model'^ and marginal rates from ratingagencies historical studies after that. For bonds longer than 15years,weixhe marginal rate at the 15-year value for that rating

category. Although there are slight kinks in the default func-tions at five years where the model and historical data meet,smoo thing techniques could be used to adjust thoserates.Sincewe think that the best estimate of marginal default rates after fiveyears are averagevalues,weuse those. Finally,apreferred methodwould be to implement the entire pricing model describedherein using stochastic default probabilities and stochastic andnegatively correlated recoveryvalues.Although we have imple-mented that methodology in other applications (Benzschaweletal. [2005]),wehave not yet explored the implications of thosemethods on our estimates ofriskydiscount rates.

E N D N O T E S'In fact, determ ination of what constitutes a credit eventcan be quite complex and a matter of some debate. In general,a credit eventisa legally defined event that includes bankruptcy,failure-to-pay, or restructuring. The ISDN and Ma rkit Partnershave devisedaprocedure whereby consenting parties may resolvethe issue of a credit event via binding arbitration (Markit Part-ners [2009]). There are many sources forabasic overview of the creditdefault swap contract. For example, see Rajan [2007].

'This assumes no premium for counterparty risk. Tycally, counterparty risk has been mitigated by requiring couterparties to post margin in response to mark-to-market loson CDS contracts. Throughout this discussion, we can cosider CDS as being traded between AA rated banks whofunding rates are LIBOR.

The Z-spread is the yield spread ofa bond referento the zero-coupon swap curve rather than the riskless zcurve usually inferred from U.S. Treasury yields.'Throughout this example, we assume that both obligare able to fund at LIBOR and that none of the securitiesquestion are trading asspecial in the repo market. Withinno-arbitrate theory, the lack of LIBOR financing and frictioin borrowing the securities in the repo market are responsib

at least in part, for the fact that there is rarely a non-zero babetween cash bonds and their corresponding C DS.'The CDSpremium is commonly, but inappropriatcalled the CD5spread based on its assumed relationship tospread to LIBOR ofits reference bond. While useful in socircumstances, calling the CDS premiumaspread obscures fact that, unlike bond coupon s of which the spread isafractof the entire cash flow, the prem ium constitutes the en tire pment fi-om the protection buyer.'The CDX.NA.IG is a basket of 125 North-Americinvestment-grade CDS contracts. A new index is issued evesix months. For a detailed description of CDS indexes, s

Markit Partners [2008].We assume for the moment that there is no cost of minating the repo(i.e.,that repo rates on the security have changed). Of course, any change in that rate would only coplicate matters as well.''In fact, those b onds might not all trade at the same cflow yield due to the different losses on all three bonds in defaFor example, assuminga40% recovery value, investor A w olose 60 points in default, whereas investors B and C would l45 and 75 points, respectively.

54 CRE DIT DEFAULT SWAPS:A CASH FLOW AS ILYSIS W INTER 2


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' Since CD S contracts trade in units of1 face of prin-cipal, it is not possible to buy protection on only 75 points.However, for any reasonably sized position, one could purchaseprotection on exactly75%of the outstanding face value of theirbond investment. O f course, we can no t directly observe physical defaultrates and must estimate those using credit ratings' historicaldefault rates or some other model-based estimate. In fact, risk-neutral default rates cannot be observed directly either, requiringail assumed recovery rates. Still, we have an additional sourceof uncertainly under the physical measure resulting from esti-mates ofphysicaldefaults.'-Estimates of physical default probabilities are criticalfor pricing u nder physical measure and ou r approach is to takehistorical cumulative default rates and modify them for creditcycle dependence using a hybrid structural/statistical defaultmodel.' 'See Corporate Default an dRecovery Rates 1920-2007.Moody's Investors Service, February 2008, and Defauh Transi-

tion an dRecovery:2007 AnnualGlobal CorporateDefault StudyandRating Transitions Standard & Poor's, February 5, 2008.'The HP13 model combinesacontingent claims approachof Merton with an Altman-like statistical approach. See Sobe-hart and Keenan [2002,2003].

REFERENCESAltman, Edward 1. Mark et Size and Investment Performanceof Defaulted Bonds and Bank Loans: 1987-2002. JOHDUI/ofApplied Finance Vol.13,No. 2 (2003),pp. 43-53 .Benzschawel, T.,L. Lorilla, and G. M cD erm ott. Effect ofSto-chastic and Correlated Defaults and Recoveries on CDOTranche Returns. Th eJournalof StructuredFinance Vol.11,No.2 (2005), pp.44-63.Benzschawel, T.,J.-H . Ryu ,J.Jiang , and M. Carnahan. Rela -tive Value among Corporate Credits. Quantitative CreditAna-lyst5, Citigroup (February 2,2006 ),pp.7-23.Bohn ,J. A Survey of Conting ent-Claim s Approaches to RiskyDebt Valuation. _/oMrMii/of Risk Finance Vol. 1,No.3(2000),pp.53-70.Clhoudhry, M. 77;eCredit Default Swap Basis. New York, NY:Bloomberg Press, 2006.Duffie, D., and K. Singleton. M ode ling Term Structures ofDefaultable Bonds. Review ofFinancial Studies Vol. 12, N o. 4(1999),pp.687-720.

Elizalde, A., S. Doctor, and Y. Saltuk. Bond-CDS Basis Hand-book.JP Morgan, 2009.Hull, J., and A. W hite. Valuing Credit D efault Swaps I: N oCounterparty Default Risk. TlteJournalof Derivatives 8(2000pp.29-40.Jarrow, R ., and S. TurnbuU. Pric ing Op tions on DerivativesSubject to Credit Risk. >i(nM/of Finance Vol.50,No .1 (1995pp.53-85.Kakodkar,A.,S.Galiani,J.G.Jonsson, andA.Gallo.CreditDeriv-ativesHandbook Vol. 1. Merr ill Lynch, 2006.King, M., and M. Sandigursky. Th e Added D imensions ofCredit; A Guide to Relative Value Trading. In Th e StructuredCreditHandbook edited by A. Rajan,G.M cDermott, and R . Ropp .111-144. New York, NY: Jo hn Wiley and Sons, 2007.Kumar P.,andS.Mithal. RelativeValuebetween Cash and DefaultSwaps in Emerging M arkets. Salomon Smith Barney,2001.Markit Partners. Mark it Credit Indices: A Primer. MarkitPartners, 2008.

. Th e CD S Big Bang: Understanding the Changes to theGlobal CDS Contract and North American Conventions.Markit Partners, 2009.O'Kane,D.Modeling Single-Name and Multi-Name CreditDeatives.Wiley, 2009.O'K ane , D., and S. Turn bull. Valuation of Cred it DefaultSwaps. Lehman Brothers,2003.Rajan, A. A Primer on Credit Default Swaps. InTlie StructureCreditHandbook edited by A. Rajan,G.McI)erniott, and R. Ropp.17-37. New York, NY: John Wiley and Sons, 2007.Sobeh art,J., andS.Keenan. Hybrid Contingent Claims Mode k:A Practical Approach to Mo deling Default Risk. InCreditRatings: Methodology Rationnte and DefaultRisk edited byM. On g, pp. 125-149. UK: Risk Books, 2002,

. Hy brid P robability of Default M odels. QuantitativeCredit Attalyst3 Cid M arkets and Banking, September 10,2003pp. 1-25.To order reprints of this article please contact DeweyPahnidpalmieri ^iijournals.com or 22-224-3675.

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