Credit Default Swaps: A Cash Flow AnalysisTERRY BENZSCHAWEL AND ALPER CORLU TERRY BENZSCHAWEL is a managing director at Citi Institutional Clients Group in New York City, NY.
[email protected] ALPER CORLU A IS an associate at Citi Institutional Clients Group 1 1 1 Now York City, NY.
[email protected] in Exhibit 1. The exhibit shows an example of a CDS written on $10 million of notional with reference to firm XYZ. The buyer of protection makes quarterly payments, the/^rc/iiii/n/ ¡eg, for as long as there is no credit event or until the maturity of the contract, whichever comes first. The CDS premium, even if trading with tection buyer, who pays the coupons or premiumsconstant coupon and upfront fee, is often (usually quarterly), and a protection seller, who expressed as an annual amount in basis points. receives the premiums, but must pay the buyer Also, contracts from a given firm are commonly the par value of an eligible security in exchange issued at a number of standard maturities, with for that security in the event of a default, bankthe most common term being five years. The ruptcy, or restructuring.' protection seller agrees to pay the buyer the face value of the CDS contract if XYZ underIn more recent versions of the standard goes a credit event and the buyer of protection CDS contract, the protecdon buyer makes an delivers to the seller the defaulted security or upfront payment set by the seller and pays a its cash equivalent." standard running 100 bp or 500 bp premium that depends on the riskiness of the reference The advantages of having a liquid credit obligor. (Even for the new contract, the upfront default swap market are well known. Prior to cash flow and fixed spread premium can be the development of the CDS market, investors converted to an effective spread premium). had few options for hedging existing credit The CDS contract is usually obligor-specific, exposures or entering a short credit position. referring to either a corporate or sovereign In such cases, investors would have to borrow entity. The securities that are eligible for bonds in an over-the-counter market, being delivery to the protection seller in event of subject to poor liquidity and high financing default, the reference obligations, are typically from costs. In addition, typical fixed-rate corporate a single class of debt (unsecured bonds, loans, bonds have huge exposure to interest-rate or subordinated bonds, etc.), but can be asset movements, which is often undesirable for specific. investors wishing to make pure credit plays. The relatively tight bid-ask spreads of CDS For most purposes, both the CDS contract contracts and, in particular, CDS index prodwith upfi-ont payment and standard coupon and ucts have provided extremely efficient means that wdth no upfront payment and market-based for investors to express views on credits of coupons can be represented using the diagram credit default swap (CDS) contract is an agreement to exchange a specified set of c o u p o n payments in return for the right to receive the par face value of a reference obligation after the particular obligor u n d e r g o e s a credit event. T h e parties involved in the contract are a pro- 40 CREDIT DEFAULT SWAPS: A CASH FLOW ANALYSIS WINTER 2011 ' That is. by enabling investors to go short with little or no initial investment. More recent contracts have an upfront cash payment to the seller of protection and either a 100 bp or . replicatitig each leg of the CDS contract itivolves several operations by both buyer and seller of protection. and options on CDS. cotwentions for trading CDS have enabled trading practices that. we suggest a pricing method based on analyses of specified and expected CDS cash flows that.EXHIBIT 1 Premium Leg Representation of a Typical CDS Contract Prior to April 2009 "P *° Maturity) Notional: US $10 MIVI C^'^ ^^''® ^ Obligor: XYZ Term: 5 Years Notional x Spread Premium (bp) Protection Buyer (Sells CDS) Protection Seller (Buys CDS) Reference Obligation ($RV) Face Value of Obligation ($100) Contingent Leg (Only Executed in Response to Credit Event) Notes: Tlie protection buyer makes regular coupon payments to the protection seller atid the contingent exchange of a reference obligation in exchange for payment of its face value in case of a credit event. Furthermore. an aspect of the CDS market that has been largely unrecognized or overlooked is that current methods for pricing and hedging CDS may be inadequate and/or problematic. respectively. In addition. Clearly. the CDS contract has provided the building block for other. For example. the no-arbitrage argument of the relationship between credit default swaps and corporate bonds states that one can replicate the premium leg of the CDS with a long position in the reference obligation combined with afixed-for-floatinginterest-rate swap and the payout leg with a short position in the reference bond atid a repo agreement to borrow that security. to the failure of somefirms. provides an alternative perspective for evaluating CDS risk and relative value. For example. the lack of a central clearinghouse for CDS trades has revealed systemic and firm-specific weaknesses in the ability to effectively manage counterparty risk. one of which implies that firms' CDS are synthetic bonds with implicit funding at LIBOR. Exhibit 2 shows the cash flows for each leg of the CDS contract in Exhibit 1 represented as an asset-swap. combined with estimates of physical default probabilities and recovery values. The no-arbitrage WINTER 2011 THEjt)URNAL OF FIXED INCOML 41 . more complicated partitioning of credit exposure via such synthetic products as single-tranche CDOs (S-CDOs). pressure from buyers of protection via CDS has been blamed for contributing directly. who can borrow the reference obligation in the repo market (Kumar and Mithal |2001 ] and Kakodkar et al.'¡OO bp running coupon but can still be represented as in the exhibit. we examine the CDS as asset-swap model atid its potential limitations. whereby investors can express a view on credit correlation with specific risk profiles. In addition. we question some assumptions tliat utiderlie the widespread application of risk-neutral pricing theory to CDS. Despite the wide success ot CDS contracts as financial instruments. CASH BOND EQUIVALENT OF CDS It is nearly axiomatic among CDS investors that a CDS contract can be replicated by long and short positions in cash bonds by the seller and buyer of protection. firms and countries around the world. whereby investors take positions on credit spread volatility.Although CDS have been widely criticized for their role in the current credit crisis. have contributed to the unprecedented volatility in cash and synthetic credit markets since mid-2007. Furthermore. recent turmoil in the credit markets have exposed vulnerabilities in the CDS market. For example. To that end. [2006]). at least in part. The asset-swap and CDS equivalence is reflected in the Z-spread. at LIBOR. the buyer can finance the bond and pass the spread premium to the protection seller. the protection buyer borrows the par amount times the notional from a bank. whichever comes first."* the common measure of adjusting cash bond spreads for comparison with CDS premiums. the buyer of protection must purchase a fiveyear bond issued by the reference obligor. assume that the bond is afixed-rateinstrument purchased at par. The remainder of the coupon. Given the general acceptance of the no-arbitrage argument between cash bonds and CDS and because we argue that this argument has limitations. call it bank Í. If there are no credit events prior to maturity of the CDS and the reference bond. is paid out as a premium to the protection seller. the interest-rate swap expires and the bond's obligor pays the face value to the protection buyer. However. the buyer of protection is due 42 CREDIT DEFAULT SWAPS: ^ CASH PLOW ANALYSts WINTER 2011 . model is vddely. we consider in detail the noarbitrage model from the perspectives of both buyers and sellers of protection. which is used to repay bank 1 for the initial loan. the obligor enters into a five-year fixed-for-floating rate swap to generate three-month LIBOR to make quarterly interest payments on the loan for the bond. assumed by most market participants. if only implicitly. To replicate this with a cash bond. The Protection Buyer A depiction of afive-yearCDS contract as an assetswap between the buyer and seller of protection appears in Exhibit 2. and by methods for estimating the cash—CDS basis as the spread to the interest-rate swap curve (Choudhry [2006]). To pay for the security.EXHIBIT 2 No-Arbitrage Model for CDS Where the Buyer of Protection Is Long a Bond Financed at LIBOR along with an Interest-Rate Swap and the Protection Seller Is Short the Reference Obligation and Has Borrowed the Bond via Repo Bank Makes 5-Year Loan Premium Leg (Paid Wfiiie No Credit Event up to Maturity) $10 MM of XYZ 5-Year Par Bond $10 MM 5-Year _Eai-Bond 5-Year Interest Rate Swap 5-Year Interest Rate Swap Notional x Spread Premium Reference Obligation ¡($RV) Fa^e Value of Obiigatioii ($100) Contingent Leg (Only Executed in Response to Credit Event) $10 MM of XYZ 5-Year Par Bond 5-Yr Swap Rate Reference Obligation Note: 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 asi set. if the reference obligor triggers a credit event prior to CDS maturity. The fact that the buyer of the bond must finance the transaction at a rate assumed to be LIBOR is the reason that the cash versus CDS basis is referenced to the bond's Z-spread. For this example. the amount above the five-year swap rate.' The coupons from the borrowed bond are used for two purposes. As long as the bond pays coupons to the buyer of protection. First. Consider first the protection buyer who agrees to make regular premium payments to the protection seller as long as there is no credit event or until the maturity of the CDS. and implementation frictions that underlie the basis. one can sell short the reference security. To replicate the payout profile of the protection seller in the cash bond market. giving rise to non-credit-related influences on both bond spreads and CDS premiums.face value minus recovery). The CDS-Cash Bond Basis The previous example is used to explain why. Assume now that the reference obligor triggers a credit event prior to maturity. The fixed leg of the swap is combined with the premium from the protection buyer to pay the coupon on the borrowed security. The protection buyer can use the payoff to deposit in bank 2 earning LIBOR and use the LIBOR proceeds to enter into a floating for fixed-rate swap until the remaining maturity of the initial fiveyear swap.but since 2006.e. The seller makes that payment from funds deposited in the bank at inception. Because of this. From the inception of the CDX index in 2003. as demonstrated in Exhibit 3. I-spread. however. Given the complexity of the relationship between a firm's reference bond and its CDS. As for the protection buyer. This way. there are a variety of market factors. technical details. in an arbitrage-free setting..g.NA. interest-rate swap agreements.the face value of the bond times the notional from the protection seller. the protection seller must pay the protection buyer the face value of the reference obligation times the notional of the CDS contract. Z-spread. deposit the sale proceeds in a bank at LIBOR. and repurchase agreements to equate cash bonds and CDS. For example.IG premiums by as much as 250 bps. Finally.. the basis has largely been negative. The importance of the present demonstration is that all of these mechanisms involve transactions in markets whose price determinants may differ from those of the deliverable obligations. borrowing arrangements. they often use the Z-spread. it should be noted that the demonstration of the no-arbitrage model of the cash bond versus CDS relation is not unique. Furthermore. the net payout will be zero and the buyer of protection withdraws the deposit in bank 2 and uses the proceeds to repay the principal on the original loan from bank 1. these factors may affect one side of the buyer/seller relationship and not the other. The Protection Seller Consider now the CDS as asset-swap from the perspective of the seller of protection as depicted in the righthand side of Exhibit 2. the model assumes that the seller of protection can borrow and lend at LIBOR and that none of the securities trade as special in the repo market. C-spread) • Imbalances between market demand for buying and selling protection • Differences in hquidity premiums for a firm's cash and synthetic assets • Impact of "cheapest to deliver" cash asset • Funding versus LIBOR • Cash reference assets trading away from par value • Difference in conventions for accrued interest on bonds and CDS premiums WINTER 2011 THE JOURNAL OF FIXED lNt:oME 43 . If a credit event is triggered prior to maturity. it is not surprising that the basis between cash bonds and CDS is rarely zero.IG)' from December 2005 through April 2009. with average bond spreads exceeding CDX.*" Despite their assumed theoretical equivalence. One could devise other combinations of bond purchases and sales. Finally. Recall that the protection seller receives quarterly payments from the protection buyer unless there is a credit event before the maturity of the CDS.NA. the basis had typically been 10 bps-20 bps positive (CDX premium greater than the average of its constituent's bond Z-spreads). the fixed-for-floadng rate swap is offset by a floating-for-fixed swap for the remaining time to maturity. and the difference between them is called the basis. at the maturity of both interest-rate swap contracts. and enter into afixed-for-floatingswap. which is a spread to the LIBOR curve. when investors wish to compare market risk premiums between CDS and their reference bonds. Z-spreads on cash bonds and their corresponding default swap spreads are rarely the same. The reference security is then delivered to the counterparty in the original short sale by the protection seller and the borrowed bond is returned as the repo is unwound whereby the seller of protection pays the lender the loss on the borrowed bond (i. The protection seller receives the reference security in exchange for the face value of the security times the notional. Exhibit 3 shows the CDS minus cash bond basis for firms in the North American investment-grade CDS index (CDX. Some of the well-known factors that influence the basis are as follows: • Method of calculating the basis (e. In fact. the break-even CDS premium should be identical to the asset-swap spread on a bond priced at par. These are discussed after the introduction of the standard method for valuing CDS in the risk-neutral setting. We list these factors only to illustrate the large number of factors that influence the CDS—cash basis. King and Sandigursky [2007]. That is. For example. the reduced-form model takes as input the market CDS premium and U. In practice. In the reduced-form approach the credit event process is modeled in continuous time as a hazard rate that represents the instantaneous probability of the firm defaulting at a particular time. for the default probabihty that results in equal expected present values of premium and contingent legs.e. as we describe in detail in the following sections. swap spreads. December 2005-May 2009 (400) Dec-05 Source: Citi and Markit Partners. [2009]). Treasury rates and solves. The contributions to the basis from the various sources that include funding rates. Jun-06 Dec-06 Jun-07 Dec-07 Jun-08 Dec-08 Jun-09 Dec-09 • 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 premiums Details of the factors underlying the cash versus CDS basis are described in detail elsewhere (Kumar and Mithal [2001]. Choudhry [2006].S. THE STANDARD MODEL FOR CDS VALUATION The standard framework for interpreting CDS values and risk management is the reduced-form model. T(i. (1) 44 CREDIT DEFAULT SWAPS: A CASH FLOW ANALYSIS WINTER 2011 . there are other serious objections to the no-arbitrage explanation for the spread relationship between CDS and their cash asset referents. unambiguously distinguishing the effects of all the factors contributing to the current basis is extremely difficult. Within the risk-neutral setting. Cash Bond Basis for Firms in the North American Investment-Grade CDS Index..T < T). A popular version of that model has been proposed by Hull and White [2000] based on the reducedform approach of Duffie and Singleton [1999| and a detailed description appears in O'Kane and Turnbull [2003]. Finally. Kakodkar et al. Then. Furthermore. in a risk-neutral setting. so are not discussed further in this article. let the time of default be denoted as Tand assume that RV is a random amount recovered if default occurs before the end of the period. is also difficult given the available market instruments. even if known. and Elizalde et al. the present value of a security is the expected value of its cashflowsdiscounted at the risk-free rate. This can be expressed in terms of the riskneutral expectations (£^) as: (r>r) e • ' " RV. and the repo market provide limitations to the utihty of the "no-arbitrage" argument as applied to credit default swaps and their cash bond asset-swaps. [2006]. hedging all the factors identified as underlying the basis. a risky zero-coupon security can be viewed as a combination of two securities: one that pays $1 at time T if the issuer does not default and one that pays RI^ if default occurs before or at maturity.EXHIBIT 3 Historical CDS vs. where RV is the recovery rate immediate after default. a CDS contract has two cash flow streams—a premium leg and a contingent leg. As noted before. expressed as percentage of the face.dng. if the firm defaults.1 can be obtained as follows: (3) •1 . the amount of contingent payment can be modeled as the face amount multiplied by (1 — Rl^. their values can be inferredfixjmmarket prices of CDS contracts. Hence the present value of the contingent leg is calculated as follows: PV conüngent (5) WlNTIiR 2011 THEJOURNAL OF FIXED INCOME 45 . On the contingent leg.Within this framework. the protecdon seller makes a single payment dependent on the occurrence of a credit event. the firm survives until the next period in which it again either defaults or survives._.R V Because the default probabilities in Equations (2)-(3) are derived in the risk-neutral set.p.RFonly if a credit event has occurred in a particular time interval with probability CPD^—CPD'^^. it has proved convenient to model default events using a Poisson counting process as introduced by Jarrow and Turnbull 11995| where the cumulative risk-neutral probability of default in the interval from 0 to t is given by EXHIBIT 4 Representation of the Contingent Leg of a Credit Default Swap in the Risk-Neutral Setting exp -¡Xju (2) where X is the instantaneous jump to default or default intensity. Because the protection buyer makes the quarterly payments of amount c/4 (where c is the annualized premium or CDS spread) conditional on the survival of the reference entity with probability 1 . 1 — RV in Exhibit 4. -+• 2 4T-1 Time in Quarters 4T In the risk-neutral pricing framework. the seller pays an amount whose value is the equivalent of the face minus the recovery value. a credit event occurs with probability p^ ^j or survives with probability 1 . The exhibit illustrates how marginal default and survival probabilities accumulate over time up undl default or maturity. For any given interval from i— 1 to f. Thus. Otherwise. the CDS spread at the initiation of the contract is assumed to reflect equal present values of the premium and the contingent legs. . and the contract terminates. usually default. Although the contingent payment is the face value of the reference bond. The premium leg consists of quarterly fixed payments made by the protection buyer to the seller until maturity or until a credit event occurs. On each time step in the exhibit. On the contingent leg. The pattern of potential contingent leg payouts of a CDS is represented in Exhibit 4.• As described previously. the present value of the premium leg equals — p PV (4) where d^ denotes the risk-free discount factor from í = 0 to t quarters and T is the maturity of the CDS in years. the probability of default by time / conditional on survival up to time í . the protection buyer must deliver the reference security or its equivalent value to the protecdon seller. whichever is first. the protection seller makes a payment 1 .CPD^. LIMITATIONS OF THE ASSET-SWAP MODEL OF CDS The traditional no-arbitrage model of the CDS as an asset-swap has proved useful for understanding the relationship between bonds and CDS and has aided the development of the CDS market. being lowest for AAA securities and highest for CCC averages. u i 0.5) and then obtain CPD'^ using c(\) and CPD^. However.2010.-•• . we can determine the values of cumulative default rates. assume that we have as input a firm's market derived CDS spreads for a range of maturities. Using Equation (6) and an assumed fraction of face value recovered in default.)*{\-RV) (6) Much of the modeling effort regarding CDS involves determining the term structure offirms'cumulative riskneutral default rates over the life of the contract.. For example..5). 2010 100 T a Q. Risk-neutral default probabilities increase monotonically with maturity for all rating classes.' -CPDl. The process continues until we obtain '^ for all maturities in the CDS term structure.Thecumularive risk-neutral probability ofdefaults. setting these two present values equal we arrive at the following expression for the CDS premium: 4 *I'Ji < ¿ . e.can then be obtained from this CDS term structure using a bootstrapping procedure: We first extract CPD^^ using c(0.g.B 10 CCC/C •AAA Source: Citi and Markit Partners. obtained in thefirststep. Finally. There are other concerns as well. These include: • potential profit or losses irom interest-rate swap positions on an asset-swap in the event of default that would not occur with a CDS. Í:(T =0.EXHIBIT 5 Cumulative Risk-Neutral Probabilities of Default by Maturity for Various Credit Rating Categories. CPD^. at each node in the lattice in Exhibit 4. 46 CREDIT DEFAULT SWAPS: A CASH FLOW ANALYSIS WINTER 2011 . i-(T= 1. March 19. .5 2 3 Maturity (years) •BBB 5 • BB ...CPD^'^. Exhibit 5 shows average cumulative risk-neutral default probabilities for March 19. i-(T = 10). *{CPD'. for credit ratings from AAA to CCC.0). large displacements in cash and CDS markets and the 300 bp fluctuation in the CDS versus cash bond basis for investment-grade credits in recent years have highlighted limitations ofthat interpretive framework. As expected. the risk-neutral default probabilities are significantly higher as credit quality decreases. approximately equal viously lower rate and the seller will have a mark-toto the yield of a BBH rated bond on June 15. At the time Bond 2 was issued. Another problem results from the fact that changes in the CDS contract may not mirror those of the underlying bonds.** Assume that. This is not an immediate issue for either consider another bond.6% 7. the buyer of protection will have unrealized in Exhibit 6. are inconsistent with the asset-swap analogy of CDS. whose coupon spread to LIBOR remains that of the original buyer. As shown in Exhibit 3. Under those circumstances.0% 85 10. • differences in the price of protection for holders of different bonds issued by the same obligor but priced in the market at different relationships to par. This has a couple of effects.4%.• failure to account for the cost of risk on the markto-market of positions in the replicating portfolio arising from changes in swap and repo rates as well as changes in either party's credit risk. the obligor to fund the LIBOR swap from the fixed bond proceeds was a much better credit and was able to issue the bond and the seller of protection will continue to receive the at par with a coupon of 3. the structure of the replicating trade will allow them to meet their contractual payoffs as if each were involved in a CDS trade. The protection seller passes the reference security to the counterparty to which it has Bond 1 Bond 2 Bond 3 (Years) 5 5 Price Coupon Yield in 100 7. consider three bonds from the swap contract.4% 7. will require less total protection since the CDS contract is written on par face value.o% 3. Assume that Bond 1 is afive-yearbond issued profit on the swap position that it has obtained at a pretoday at par with a coupon of 7%. a new buyer would have to make a lower absolute payment to fund that same bond at that lower price. an investor wishing to enter into a CDS contract on that bond will have to buy protection on less than full face value of the bond.2009.0% r7. all ñx)m the same issuer whose indicative information appears else equal.0% 115 Default 40% 40% 40% sold the bond short and terminates the repo (or arranges a reverse repo). one might expect the CDS price to change as marginal buyers of protection for bonds bought at less than par. Failure of the Law of One CDS Price investors in the replicating long. prior to default. Profitability of Asset-Swaps in a CDS Replicating Portfolio Consider first the mark-to-market sensitivities of a buyer and seller of protection that enter into each side of an asset-swap replication of a CDS contract as depicted in Exhibit 2. EXHIBIT 6 the investor in an asset-swap is exposed to the CDS Indicative Data for Three Five-Year Senior Unsecured versus cash basis. changes in the Bonds at Par. which almost always trade at premiums to LIBOR. First. For example. That is. Bond 2. originally issued with a investor. • consistent positive CDS premiums for AAA rated credits. if swap rates rise. Discount. For example. However. For such investors. the borrower of the bond is now paying a higher rate than LIBOR on the face value of the bond at its current market price. even assuming that the Another issue with the CDS as asset-swap analogy reference bond is trading at par and each side is able to is that the value of credit protection on a single obligor borrow at LIBOR. For example. and Premium Prices Relative to Par basis can be large and have moved as much as several hundred basis points over several months. the price of the fixed-rate borrowed bond will have decreased from par value due to an overall rise in rates that is independent of its credit quality. as demonstrated in the following section. because the protection buyer must continue as Bond 1. That is. Recovery Maturity Consider also what happens if the bond issuer in an asset-swap defaults.and short-CDS portfolios will have additional exposures. Presumably. However. swap rates have increased. Furthermore. Now market loss. aside from potential credit exposure and/or 15-year maturity exactly 10 years ago by the same issuer margin calls. The seller of protection must deliver the notional times the par face value of the reference obligation to the protection buyer in return for the reference security. the buyer and seller of will differ for investors of its bonds with different coupons protection will be exposed to movements in swap rates at the same maturity. 5 WINTER 2011 THE JOURNAL OF FIXED INCOME 47 . now that the increased LIBOR from their bank. . Since CDS are quoted in units of 100 points of face value.5-year par bond to the value of 100 when discounted by the corresponding Treasury yield.^ curve for the BBB rated par bonds on June 15. June 15.2009. CPD^^.S. Since both Bond 1 and Bond 2. so what is the correct "fair" price of a CDS protection on an obligor having issued Bonds 1 and 2? Clearly. respectively. Thus. The risk-neutral cumulative probabihty of default. Treasury discount curves is 414 bp per annum. Assume that we bought Bond 2 on June 15. Treasury yield curve. we might require less default protection if we owned Bond 2 than if we owned Bond 1. We determine the necessary premium on a CDS for default protection on 100 face of Bond 2." = 275 bp ^ 2 3 Maturity (years) CDS contracts are quoted in units of 100 face regardless of the prices of bonds eligible for delivery in default. To break even in default. Now consider Bond 2. the need for less protection on Bond 2 should translate into fewer CDS contracts for a given notional amount of bonds than for Bond 1 (or an equal number of contracts at a lower spread premium). CPD'^ . when borrowing by B rated credits required a coupon of 10. Within the CDS as asset-swap model. Thus. Bond 2 was issued 10 years ago as a 15-year bond at 3. will have a 40% recovery of face value in default. Because Bonds 1 and 2 are from the same obligor. 2009. we assume a recovery value (in this case 40% of principal) and solve for the CPD^^ that equates the cashflowsfromthe 0. this should be equal to N.RK 10045 =— Xf ^Q p. c. The resulting value off is 275 bps per annum.S. equating these two values. 2009. As mentioned. we have 85-40 = N .4% and now has 5 years to maturity. We can approximate this premium as a yield spread to LIBOR by subtracting the difference between LIBOR and Treasury spot rates on June 15. consider a third bond issued five years ago at par. * 1 RV 100 (7) EXHIBIT 7 Risk-Neutral Default Probabilities as a Function of Maturity Implied by BBB Bonds and U.obligor is BBB rated. while trading at 100 and 85.15. * (1 . At the current coupon rate of 7%.6%. we can solve for the premium. a price well below the par value of Bond 1. For example. for the example of Bonds 1 and 2. at any maturity for this obligor can be calculated from the obligor's par yield curve and the U. That is. investors are demanding a 7% yield and Bond 2 is trading at a discounted price of 85. In terms of CDS contract. This process is repeated at regular intervals over the life of the bond in question. the discount bond issued by the same obligor as Bond 1. The value for the premium obtained using U.S. analysis based on the risk-neutral credit curve would suggest two different prices for CDS protection per 100 notional of Bonds 1 and 2. The resulting CPD'. we can use the values for CPD^ at various maturities in Exhibit 7 to calculate the expected premium of a credit default swap via Equation (6). that bond is now trading at 115. a difference of 48 bps. In fact. using the same risk-neutral default probabilities and recovery rate that we used for Bond 1. 366 bps per annum on Bond 1 and 48 CREDIT DEFAULT SWAPS: A CASH FLOW ANALYSIS WINTER 2011 . 85 — 40) from the protection seller. Treasury Par Yield Curves.5-year par bond. at 85. appears in Exhibit 7.^. 8 5 . 2009 and therefore N^ . our estimate of a five-year CDS premium for this BBB obligor is roughly 366 bps.RV/ÍÜÚ). Finally. to find the risk-neutral default probability for the 0.e. a simple way to determine the CDS premium adjustment required for a bond trading away from par is to realize that the relative premium between a par and non-par bond from the same firm is directly proportional to their losses in default. the buyer of Bond 2 will need to get 45 points per 100 face (i. Let N. be the notional amount of protection that we need to buy to neutralize the default risk of Bond 2. B. and C each buy CDS on 100 units of face and pay $3. our analysis suggests that the price of CDS protection necessary to recover the price in default is 458 bps.7% 38.'' Suppose each investor buys protection on their respective investments to cover potential losses from default and. investor B will have windfall gains of 39%. However. assume that the cost of CDS protection on 100 face is 3.7% if default occurs within the first six months.7% 2. IRRs under historical default rates for investors A and B are similar. buying protection on 100 face of CDS. B owns 100 face of Bond 2.e. That is.1% 3. owing to the 15 points of EXHIBIT 8 Returns from Investing in Hypothetical BBB Rated Par and Non-Par Bonds for CDS Protection Bought for Full Face Value of Bonds or Adjusted for Loss Given Default IRR CDS Spread (Historical per Unit Default Rates) Bond Face 366 366 275 366 458 3. Investor A will pay $100 for Bond 1.4% under average default conditions but a return of only 1.4% -22.. Case 2: Investor A again purchases protection of 100 units of face at $3. this advantage for investor A will only result on average. investor C will receive $125 from the CDS contract but must deliver an additional 25 units of face at a recovery value of $40 into the CDS contract at a cost of $10. In each scenario.'" In default.4%. Exhibit 8 displays the internal rates of return (IRRs) for investors A. Similarly.275 bps per annum on Bond 2. However. thereby providing potential arbitrage opportunities. The 0. B. In Case 1.6% coupon and trading at 115 for 100 notional. and C. and C under two different default scenarios for each of the two hedging scenarios. Consider for comparison the returns for investor B.28% (i. the annual historical rate for a BBB credit) over the five-year contract term. whereas investor B pays only $85 for Bond 2 and investor C pays $115 for Bond 3.66.9% IRR (Default in the First 6 Months) 1. as shown in the far right column of Exhibit 8.3%i advantage for investor A in C'ase 1 results from the larger coupon on the par bond and occurs despite the 15-point advantage in cases of default for investor B. Case I: Investors A. investor B now buys CDS protection on only 75% of the outstanding face of the bond purchased at $85.66 a year until maturity or default. thereby recovering their investment. consider three investors. for a firm having bonds with different market prices relative to par. we assume a constant default rate of 0.66% per annum. Now. and C owns 100 face of Bond 3. and C. That is.8% 0. consider expected returns from two different hedging scenarios involving investors A. for Bond 3 fi-om this same obligor. if the issuer defaults in the first six months. The first row in Exhibit 8 shows that investor A has an expected IRR of 3. WINTER 2011 THE JOURNAL OK FIXED INCOME 49 . B.e. in defluilt. having paid $ 115 for 100 face of bond.. assuming that the bonds all trade at equivalent cash flow yields.7% Investor A B(Case 1) B (Case 2) C (Case 1) C (Case 2) Bond Type Par Discount Discount Premium Premium Face of CDS 100 100 75 100 125 Note: The average cumulative five-year default rate for a BBB rated credit is roughly 1.1% 4. where A owns 100 face of Bond 1 in Exhibit 6. One implication of the foregoing analysis is that. having a 10. B. A. based on the risk-neutral default curve in Exhibit 7. B wiU receive $75 from the CDS contract and the recovery value of $40 on the excess 25 points of face not delivered into the contract (i. For example.4% 3. $10). buys 125 units of CDS protection to cover potential loss of investment from default. Investor C.0% 2. the cost of protecting one's original principal can vary significantly. 25 0. receives little benefit from the pull to par at maturity. The foregoing analysis has several implications for investors in bonds and CDS. In fact. exceeding investor A's IRR of 3. The advantage in expected returns to investor B under historical default rates results from a combination of relatively low default rates and the "puU-to-par" at maturity under conditions of no default. it is not clear which hedging strategy is best for nonpar securities as each has its advantages and disadvantages. The values of cpd¡ are expected cumulative probabilities of actual defaults from time f = 0 to t — !. however. we can assume that we know precisely the term EXHIBIT 9 Example of Quarterly Cash Flows from Premium Leg (left-hand panel) and Contingent Leg (right-hand panel) . In order to determine the size of the payouts in default one must assume a recovery value for the reference security. investor B buys CDS protecdon on only 75% of the outstanding face. In Case 2. That is because the larger payouts are weighted by the probabilities of default.cpdj) 0 0. However.75 1 0 0. while having his investment protected. these examples reveal clear limitations of the CDS as assetswap analogy as a general framework for relating bond prices to CDS spreads. In this case. 50 CREDIT DEFAULT SWAPS: A CASH FLOW ANALYSIS WINTER 2011 .1%. B's IRR will be only 0.5 (years) 0. For example. when valuing the CDS under physical measure we must also assume a term structure of physical default rates. Investor B's annual premium payments therefore will be only $2. Exhibit 9 displays expected cash flows from premium and contingent legs of a hypothetical oneyear CDS contract. investor C has a greater IRR than investor A and investor B in Case 2.excess return on the 100 face of CDS protection. However. More importantly.4%. whereas investing in premium bonds has the greatest downside risk." Notice that the size of the premium payments shown on the left-hand side in Exhibit 9 decreases over time. for an early default. Buying and hedging discount bonds appears to be the best overall strategy. That is.25 0. ANALYSIS OF CDS CASH FLOWS Civen the limitations of the CDS as asset-swap framework described in the previous secdon. for average default scenarios.4%. Over any quarterly interval. it is clear that the risk-reward aspects of CDS protection will differ for investors in par and non-par securidesfixjmthe same issuer. investor C performs slighdy better than investor A (and B).9%. reflecting the fact that their probabilities of payment decrease as the likelihood of survival decreases over time. expected average CDS payouts in default are typically smaller than their associated premiums. is lowest at 2. In addition.Year CDS Contract (cashflows not drawn to scale). The contingent payments appear on the right-hand side of Exhibit 9. but for early default. First. For now. The expected IRR for investor C in Case 2. owing to the large premium required for 125 points of CDS protection. investor C suffers a huge loss of 22% on the initial $115 investment. because B. Exhibit 8 displays IRRs for investor C in Case 1 that are the reverse of those for investor B. the expected IRR for investor B increases to 4. should a default occur in the first six months. owner of the premium bond. In the proposed cashflow framework. we value the premium and condngent legs as sums of the discounted values of their expected cash fiows under physical measure.75 per unit face per annum per unit of bond face owned. we introduce a simplified approach to evaluating CDS contracts and explore its usefulness for interpreting market data on CDS spreads.75 Aofe: L is the Loss Given Default of a Hypothetical One.5 (years) 0. as shown in the bottom panel. obtained during the high-liquidity. tight spread environment of March 2006. who are "long" credit exposure. market CDS premiums also vary greatly over the credit cycle. Notice that default curves are ordered by ratings and show only slight adjustments with changes in the credit cycle.structure of physical default rates for all tenors for all credit rating categories. we use market values of CDS premiums to infer the excess compensation required by sellers of protection. particularly during periods of market stress as shown in the middle table of Exhibit 11.) To calculate breakeven CDS premiums. If correct. the CDS spreads in the middle panel near the height of the crisis are nearly a factor of 10 larger than those three years earlier. we need estimates of physical default probabilities such as those shown in the left-hand panels of Exhibit 10 for several points in the credit cycle. some of these negative values are a sizable fraction of their estimated breakeven spread premiums. CDS spreads in the top lefthand panel. That is. However.j. with values in the middle panels taken at cyclical wide spreads. the risk premiums for high-quality names have been negative. spread premiums above those required to compensate for default can be quite large relative to breakeven spreads. h¡. like breakeven CDS spreads. We will consider the implications of estimating those probabilities in a later section. Notice that. More generally. pre-crisis period. these negative risk premiums suggest that. We can compare market CDS spreads with the inferred breakeven spreads to determine CDS risk premiums above the calculated compensation for default.'.)*{\-RV) (10) Unlike in the risk-neutral setting there is. for relatively high-quality credits at short tenors. since CDS spreads in the assetswap framework are thought to be spreads over LIBOR rather than simple cashflows. the excess CDS spreads over breakeven values) at the corresponding points in the credit cycle. it is possible that the relatively small negative risk premiums reflect errors in our assumed default probabilities and/or recovery values. Default rates are obtained from historical data and combined with values obtained from a market-based Merton-type model as described in the Appendix. Each plot shows cumulative default curves by rating. for their large promised payouts in the event of default. (11) (Note that within the asset-swap framework. although most values for the risk premium in the tables are positive. no expected premium that relates the PVs for the contingent and premium legs. h. For example. Exhibit 11 displays CDS spreads by tenor and rating category for the same three points in the credit cycle as in Exhibit 10. as the crisis abated in 2010. Notice that.adding LIBOR to the CDS WINTER 20 THE JOURNAL OF FIXED INCOME 51 . To can calculate the present values of the premium and contingent legs under physical measure using similar equations as those for risk-neutral pricing in Equations (4) and (5). Our approach is to first calculate a breakeven premium.The physical CPD values at the left in Exhibit 10 are used along with the assumption of an average 40% recovery value in default to generate the breakeven CDS premium curves in the corresponding right panels of the exhibit.e. In contrast. spreads have returned to roughly their pre-crisis levels.That is. It is important to note that the inferred negative risk premiums in the top and bottom tables of Exhibit 11 would likely not be evident within the CDS as bond assetswap framework. during periods of relatively low spread levels. a priori. breakeven spread premiums required by sellers of protection vary by more than a factor of 10 over the cycle. except we substitute cpd^ for CPD''' such that 4i\' PV p (9) and 4N PV onungciit c *{cpd -cpd . Of course.000 bps. as the annualized quarterly premium necessary to equate the present values of the contingent and premium legs as calculated using expected physical cashflows.. That is. sellers of protection during periods of high liquidity did not receive sufficient compensation to cover their average expected payouts in default. However. The top plots are for the high liquidity. are relatively tight with even CCC spreads below 1. The tables in the right-hand portion of Exhibit 11 display by rating and tenor the CDS risk premiums (i. and those in the lower graphs from a more average spread and default environment. would correspond to the CDS premium. Finally. However. We find limitations of the bond-CDS equivalent hypothesis including difficulties in replicating a bond synthetically in the CDS and repo markets and different implicit CDS premiums for an obligor's bonds of the same maturity but trading at different prices relative to par.01 2 3 4 5 6 7 8 9 10 Maturity (years) AAA AA A BBB S m ra BB B CCC/C Note: Breakeven CDS premiums are calculated using Equation (6) and the corresponding default probabilities and a recovery rate in default of 40%. it would seem difficult to escape the conclusion that in these times of high liquidity and tight credit spreads. premium would result in a greater spread than those shown for breakeven. sellers of CDS protection have been undercompensated for their risk. n •a ra (0 V) 1000 100 10 1-Mar-10 1 1 Maturity (years) 10 I c O) I 0. 1000 : 100 10 CD 1-Mar-06 'S. I 2 „ 10000 a. The implied cash flows from each leg under physical measure were then discounted at appropriate risk-free rates and the resulting breakeven premiums were compared to the market-implied CDS spreads. Results were presented for various rating categories and tenors for several points in 52 CREDIT DEFAULT SWAPS: A CASH FLOW ANALYSIS WINTER 2011 . We also examined an alternative approach to CDS valuation.EXHIBIT 10 Estimates of Physical Cumulative Default Probabilities (left-hand side) and Breakeven CDS Premiums (right-hand side) by Credit Rating and Tenor for Three Different Points in the Credit Cycle 100 •s 10000 'S a. 10000 lA •o s a 1000 100 .>. Source: Citi and Moody's Investors Service. one that uses assumed physical default rates as opposed to risk-neutral default rates. SUMMARY AND IMPLICATIONS We examined the assumption that credit default swap (CDS) contracts can be replicated by bonds funded at LIBOR along with an interest-rate swap and a repurchase agreement. BB ....EXHIBIT 11 CDS Spreads and Difference between CDS Spreads and Breakeven Spreads by Rating Category and Tenor for Three Different Points in the Recent Credit Cycle 10000 Market Premiums Minus Breakeven CDS Spreads March 1.. by knowing the agency WINTER 2011 THE JOURNAL OF FixEn INCOME 53 . but data are available for out to 30 years. Fortunately.. the credit cycle. APPENDIX ESTIMATING PHYSICAL DEFAULT PROBABILITIES A critical assumption underlying the proposed methodology is that cumulative default functions over time are known for all credits. A summary of those ratings appears in Exhibit Al. analyses of CDS cashflowsunder physical measure highlight asymmetries between risk premiums received by investors in bonds versus CDS that are not evident from similar comparisons within the CDS as asset-swap framework. Furthermore. Cumulative rates by year are shown out to 15 years.B —-A CCC/C BBB BBB BB B 457 1244 1191 1094 5665 4897 4544 4409 March 1. The methodology enables measurement of the minimum amount of risk premium compensation that must be required by the protection seller to break even for expected default." Thus. such as Moody s and Standard & Poor's. the credit rating agencies. Our analyses reveal that. at times. 2009 75 71 67 136 389 5 75 71 77 153 381 7 10 83 84 75 80 90 101 162 1 169 366 373 1075 994 CCC/C Maturity AAA AA A 10 Maturity (years) AAA .2006 Maturity 1 3 7 5 10 AAA 1 3 7 12 17 -1 AA -11 -8 -2 5 1 A 0 14 -20 -11 BBB BB B 5 35 18 546 1 112 109 111 194 -18 18 35 609 3 -4 61 98 707 10 87 125 705 25 108 155 700 CCC/C Maturity AAA AA A BBB BB B March 2. have compiled extensive statistics on cumulative default rates for issues with given initial ratings.AA . Inc. market CDS premiums for high-quality credits were often insufficient to compensate sellers of protection for expected payouts from default.2010 1 17 15 15 26 75 241 619 3 -1 -28 -24 -10 81 362 767 5 7 -10 -7 12 141 436 891 7 18 3 9 3478 10 32 20 30 156 442 847 26 48 168 443 788 CCC/C Source: Citi and Markit Partners. whereas investors B and C would lose 45 and 75 points.6 2.e.8 1. that repo rates on the security have not changed).3 1.0 14. in practice.0 0.2 21. ''In fact. one can derive the probabilities of default for each successive time period.5 0.2 63. unlike bond coupons of which the spread is a fraction of the entire cash flow.1 0.9 24.2 28. Finally. "We assume for the moment that there is no cost of terminating the repo (i.5 5. For a detailed description of CDS indexes. failure-to-pay. Typically. ranging from over 12% in some years to less than 1% in others (Altman [2003]).ILYSIS WINTER 2011 .7 1.7 27. a preferred method would be to implement the entire pricing model described herein using stochastic default probabilities and stochastic and negatively correlated recovery values. While useful in some circumstances. we can consider CDS as being traded between AA rated banks whose funding rates are LIBOR.3 2.1 2.2 1.6 6.5 0.7 4.0 0.3 1.4 1. For example. credit rating (or an analyst's or model equivalent rating).1 45.9 42. the lack of LIBOR financing and frictions in borrowing the securities in the repo market are responsible. at least in part. we use those.6 54.6 49.2 0. calling the CDS premium a spread obscures the fact that. 'The CDS premium is commonly. "There are many sources for a basic overview of the credit default swap contract.5 58.8 39.9 nn Source: Data are from Moody's Investors Service.0 35. smoothing techniques could be used to adjust those rates. the premium constitutes the entire payment fi-om the protection buyer. for the fact that there is rarely a non-zero basis between cash bonds and their corresponding CDS. Treasury yields.0 1.5 66.3 22. Since we think that the best estimate of marginal default rates after five years are average values. we have not yet explored the implications of those methods on our estimates of risky discount rates. wefixthe marginal rate at the 15-year value for that rating category. but inappropriately. 54 CREDIT DEFAULT SWAPS: A CASH FLOW AS. For example.0 0.4 9.7 8.5 9.6 19.0 1. Exhibit Al shows default probabilities out to 15 years constructed as a combination of marginal default rates in years 1 — 5 from Sobehart and Keenan's Hybrid Probability of Default (HPD) model'^ and marginal rates from rating agencies historical studies after that. Although we have implemented that methodology in other applications (Benzschawel et al. Throughout this discussion.0 37.5 50. For bonds longer than 15 years.9 1.2 44.3 0.2 66. 'The CDX. Of course. assuming a 40% recovery value.3 7. we use a Mertonbased contingent claims analysis model to derive estimates of default probabilities from one to five years and historical values after that.8 47.1 0. 'This assumes no premium for counterparty risk.5 59. Although there are slight kinks in the default functions at five years where the model and historical data meet.2 1.5 66.5 67.6 26.2 2. [2005]).9 3.3 5.0 7. a credit event is a legally defined event that includes bankruptcy. Within the no-arbitrate theory.2 45.NA.EXHIBIT Al Cumulative Default Probabilities for Bonds by Rating Category as a Function of Years since Issuance. In general.7 2.5 66. determination of what constitutes a credit event can be quite complex and a matter of some debate.3 49.5 12.2 0.6 1.2 3. called the CD5 spread based on its assumed relationship to the spread to LIBOR of its reference bond. "•The Z-spread is the yield spread of a bond referenced to the zero-coupon swap curve rather than the riskless zero curve usually inferred from U. see Markit Partners [2008].0 5.5 2. A new index is issued every six months.5 29.9 57.2 0.4 25. investor A would lose 60 points in default.6 0.5 2. or restructuring. For example.3 0.S.8 0.7 39.4 0. 1920-2006 1 AAA AA A BBB BB B CCC 2 3 4 5 6 7 8 9 10 11 12 13 14 15 07 1.7 17. those bonds might not all trade at the same cash flow yield due to the different losses on all three bonds in default.3 2.6 32. One problem with using historical default rates for determining default probabilities is that default rates are credit-cycledependent. we assume that both obligors are able to fund at LIBOR and that none of the securities in question are trading as special in the repo market.9 10.5 6.9 67. see Rajan [2007]. Thus.IG is a basket of 125 North-American investment-grade CDS contracts. The ISDN and Markit Partners have devised a procedure whereby consenting parties may resolve the issue of a credit event via binding arbitration (Markit Partners [2009]).1 24. any change in that rate would only complicate matters as well..9 00 ni 0 1 0 ? 0 4 On OR 0 7 0 7 0 7 0 7 0 7 0 7 0.0 19. ENDNOTES 'In fact. respectively.4 12. counterparty risk has been mitigated by requiring counterparties to post margin in response to mark-to-market losses on CDS contracts. 'Throughout this example.5 28. In fact. Lorilla.. Hull. and M. "Relapp. February 5.'"Since CDS contracts trade in units of 100 face of principal. 43-53. M. NY: Credit Attalyst 3.e Credit Default Swap Basis.J. Benzschawel.. pp. we have an additional source of uncertainly under the physical measure resulting from estimates of physical defaults. edited by A.. Still. Jarrow. "Markit Credit Indices: A Primer. Singleton. 53-85. 2007. 7-23. No. 1. REFERENCES Altman. McDermott. Vol. 1. and M." Salomon Smith Barney. . "Market Size and Investment Performance of Defaulted Bonds and Bank Loans: 1987-2002.S. NY: John Wiley and Sons. 11. New York. 2009. Mithal. 687-720. TurnbuU. "Pricing Options on Derivatives Subject to Credit Risk.2003]. Doctor. 13. "Valuation of Credit Default chastic and Correlated Defaults and Recoveries on CDO Swaps.Jonsson.. Kumar. 17-37.J. However. NY: John Wiley and Sons. pp. pp." In The Structured Credit Handbook. 2007. 29-40. Vol. 2003. Wiley.2006).. Rating Transitions. Vol. Gallo. February 2008. No."_/oMrMii/ of Risk Finance. Credit Derivatives Handbook. 1920-2007. Vol. please contact Dewey Pahnieri at dpalmieri(^iijournals. "Modeling Term Structures of Defaultable Bonds.">i(nM/ of Finance. 2002.2003." Markit Partners. and K.J. JP Morgan. 2008. O'Kane..J. Rajan. 12. 4 (1999). and S. 2 (2003). Ryu. Moody's Investors Service. "Effect of StoO'Kane.. September 10. Standard & Poor's. UK: Risk Books. pp. To order reprints of this article. and G. WINTER 2011 THE JOURNAL DH FI. L. See Sobehart and Keenan [2002." In Credit Bohn. Rationnte and Default Risk. Vol. Keenan.." Tlte Journal of Derivatives.. pp. "Hybrid Probability of Default Models." JOHDUI/ of Applied Finance. 2 (2005). pp. 53-70.com or 2Í2-224-3675. Bond-CDS Basis Handbook. 3 (2000).. 2001. No. Cid Markets and Banking. T. Saltuk." Quantitative Credit Analyst 5. Citigroup (February 2. and R. Rajan. "A Survey of Contingent-Claims Approaches to Risky Ratings: Methodology. Merrill Lynch. P. it is not possible to buy protection on only 75 points. 2009. and S. '••The HP13 model combines a contingent claims approach of Merton with an Altman-like statistical approach. pp. 50. 2009. New York. 44-63. . Roy. Transition and Recovery: 2007 Annual Global Corporate Default Study and Elizalde. S. R. Bloomberg Press.. tive Value among Corporate Credits. and A." Markit Partners. 125-149. "A Primer on Credit Default Swaps. Sandigursky.J. A Guide to Relative Value Trading. M. Markit Partners. Roy.-H. T. D. and Defauh. Duffie. edited by A. "Hybrid Contingent Claims Modek: A Practical Approach to Modeling Default Risk. Kakodkar. pp." Review of Financial Studies." The Journal of Structured Finance. Ong. G. 111-144. "The Added Dimensions of Credit. Edward 1. pp. A." Quantitative Clhoudhry. New York. Vol. Benzschawel. 2006. M. and A. No. Carnahan. 2006. McI )erniott. edited by Debt Valuation." Lehman Brothers. A. "The CDS Big Bang: Understanding the Changes to the Global CDS Contract and North American Conventions. 2008. A. Turnbull. Galiani. J. D. "Valuing Credit Default Swaps I: No Counterparty Default Risk. and R. requiring ail assumed recovery rates.G. '-Estimates of physical default probabilities are critical for pricing under physical measure and our approach is to take historical cumulative default rates and modify them for credit cycle dependence using a hybrid structural/statistical default model. D. and S. McDermott. 1 (1995). 8 (2000).XED INCÍOME 55 . "Relative Value between Cash and Default Swaps in Emerging Markets.Jiang. ''See Corporate Default and Recovery Rates. Tranche Returns. Sobehart. and S. White. for any reasonably sized position. one could purchase protection on exactly 75% of the outstanding face value of their bond investment. G. 77. riskneutral default rates cannot be observed directly either. pp. we can not directly observe physical default rates and must estimate those using credit ratings' historical default rates or some other model-based estimate. Rajan. and Y. Modeling Single-Name and Multi-Name Credit Derivatives. "Of course." In Tlie Structured Credit Handbook. No. 1-25. King. asp . This material must be used for the customer's internal business use only and a maximum of ten (10) hard copy print-outs may be made.©Euromoney Institutional Investor PLC.iijournals. Source: Journal of Fixed Income and http://www.com/JFI/Default. No further copying or transmission of this material is allowed without the express permission of Euromoney Institutional Investor PLC. 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