1 VALUING CREDIT DEFAULT SWAPS I: NO COUNTERPARTY DEFAULT RISK John Hull and Alan White Joseph L. Rotman School of Management University of Toronto 105 St George Street Toronto, Ontario M5S 3E6 Canada Tel: (416) 978 8615 (Hull) Tel: (416) 978 3689 (White) April, 2000 1

2 VALUING CREDIT DEFAULT SWAPS I: NO COUNTERPARTY DEFAULT RISK Abstract This paper provides a methodology for valuing credit default swaps when the payoff is contingent on default by a single reference entity and there is no counterparty default risk. The paper tests the sensitivity of credit default swap valuations to assumptions about the expected recovery rate. It also tests whether approximate no-arbitrage arguments give accurate valuations and provides an example of the application of the methodology to real Valuing Credit Default Swaps II: Modeling Default data. In a companion paper entitled Correlation , the analysis is extended to cover situations where the payoff is contingent on default by multiple reference entities and situations where there is counterparty default risk. 2

3 Credit default swaps have become increasingly popular in recent years. Their purpose is to allow credit risks to be traded and managed in much the same way as market risks. In 1998, trading in credit default swaps was facilitated by standard documentation produced by the International Swaps and Derivatives Association. A credit default swap (CDS) is a contract that provides insurance against the risk of reference entity and a a default by particular company. The company is known as the default by the company is known as a credit event . The buyer of the insurance obtains the right to sell a particular bond issued by the company for its par value when a credit reference obligation and the total par value of event occurs. The bond is known as the notional principal . the bond that can be sold is known as the swap’s The buyer of the CDS makes periodic payments to the seller until the end of the life of the CDS or until a credit event occurs. A credit event usually requires a final accrual payment by the buyer. The swap is then settled by either physical delivery or in cash. If the terms of the swap require physical delivery, the swap buyer delivers the bonds to the seller in exchange for their par value. When there is cash settlement, the calculation Q , of the reference obligation some agent polls dealers to determine the mid-market price, − Q )% specified number of days after the credit event. The cash settlement is then (100 of the notional principal. An example may help to illustrate how a typical deal is structured. Suppose that two parties enter into a five-year credit default swap on March 1, 2000. Assume that the notional principal is $100 million and the buyer agrees to pay 90 basis points annually for protection against default by the reference entity. If the reference entity does not default (that is, there is no credit event), the buyer receives no payoff and pays $900,000 on March 1 of each of the years 2001, 2002, 2003, 2004, and 2005. If there is a credit event a substantial payoff is likely. Suppose that the buyer notifies the seller of a credit event on September 1, 2003 (half way through the fourth year). If the contract specifies physical settlement, the buyer has the right to sell $100 million par value of the reference obligation for $100 3

4 million. If the contract requires cash settlement, the calculation agent would poll dealers to determine the mid-market value of the reference obligation a predesignated number of days after the credit event. If the value of the reference obligation proved to be $35 per $100 of par value, the cash payoff would be $65 million. In the case of either physical or cash settlement, the buyer would be required to pay to the seller the amount of the annual payment accrued between March 1, 2003 and September 1, 2003 (approximately $450,000), but no further payments would be required. binary There are a number of variations on the standard credit default swap. In a , the payoff in the event of a default is a specific dollar amount. In a credit default swap basket credit default swap , a group of reference entities are specified and there is a payoff when the first of these reference entities defaults. In a contingent credit default swap , the payoff requires both a credit event and an additional trigger. The additional trigger might be a credit event with respect to another reference entity or a specified movement in some , the notional amount determining the dynamic credit default swap market variable. In a payoff is linked to the mark-to-market value of a portfolio of swaps. In this paper we explain how a plain vanilla and binary credit default swap can be val- ued assuming no counterparty default risk. Like most other approaches, ours assumes that default probabilities, interest rates, and recovery rates are independent. Unfortunately, it does not seem to be possible to relax these assumptions without a considerably more complex model. However, we are able to reach some general conclusions about the impact of the assumptions on CDS valuations. We test the sensitivity of our valuations to assumptions about the amount claimed in the event of a default and the expected recovery rate. We also test whether approxi- mate no-arbitrage arguments give accurate valuations. In a later paper, Hull and White (2000), we will explain how the analysis can be extended to cover situations where the payoff is contingent on default by multiple reference entities and situations where there is counterparty default risk. 4

5 1. Estimation of Default Probabilities The valuation of a credit default swap requires estimates of the risk-neutral probability that the reference entity will default at different future times. The prices of bonds issued by the reference entity provide the main source of data for the estimation. If we assume that the only reason a corporate bond sells for less than a similar Treasury bond is the possibility of default, it follows that: − Value of Corporate Bond = Present Value of Cost of Defaults Value of Treasury Bond By using this relationship to calculate the present value of the cost of defaults on a range of different bonds issued by the reference entity, and making an assumption about recovery rates, we can estimate the probability of the corporation defaulting at different future 1 If the reference entity has issued relatively few actively traded bonds, we can use times. bonds issued by another corporation that is considered to have the same risk of default as the reference entity. This is likely to be a corporation whose bonds have the same credit rating as those of the reference entity—and ideally a corporation in the same industry as the reference entity. We start with a simple example. Suppose that a five-year zero-coupon Treasury bond with a face value of 100 yields 5% and a similar five-year zero-coupon bond issued by a corporation yields 5.5%. (Both rates are expressed with continuous compounding.) The × − 0 . 05 5 or 77.8801 and the value of the corporate bond e value of the Treasury bond is 100 − × 0 . 5 055 e . 9572. The present value of the cost of defaults is, therefore is 100 =75 77 . 8801 − 75 . 9572 = 1 . 9229 Define the risk-neutral probability of default during the five-year life of the bond as p . If we make the simplifying assumption that there are no recoveries in the event of a 1 We assume that Treasury rates are the benchmark zero-default-risk rates. Some an- alysts argue that, because of the tax treatment of Treasury bonds in the United States and other issues, it may be more appropriate to use Agency rates or LIBOR rates as the benchmark. Our analysis can be adjusted to do this. 5

6 default, the impact of a default is to create a loss of 100 at the end of the five years. The expected loss from defaults in a risk-neutral world is, therefore, 100 and the present value p of the expected loss is 0 . 05 × 5 − pe 100 It follows that: 5 − 0 . 05 × pe 100 . 9229 =1 p . 0247 or 2.47%. so that =0 There are two reasons why the calculations for extracting default probabilities from bond prices are, in practice, usually more complicated than this. First, the recovery rate is usually non-zero. Second, most corporate bonds are not zero-coupon bonds. When the recovery rate is non-zero, it is necessary to make an assumption about the claim made by bondholders in the event of default. Jarrow and Turnbull (1995) and Hull and White (1995) assume that the claim equals the no-default of the bond. Duffie and Singleton (1997) assume that the claim is equal to the value of the bond immediately prior to default. As pointed out by J.P. Morgan (1999) and Jarrow and Turnbull (2000), these assumptions do not correspond to the way bankruptcy laws work in most countries. The best assumption is that the claim made in the event of a default equals the face value of the bond plus accrued interest. is t As mentioned earlier, the payoff from a CDS in the event of a default at time t . usually the face value of the reference obligation minus its market value just after time Using the best claim amount assumption just mentioned, the market value of the reference obligation just after default is the recovery rate times the sum of its face value and accrued interest. This means that the payoff from a typical CDS is L − RL [1 + A ( t )] = L [1 − R − A ( t )] (1) ( where is the notional principal, R is the recovery rate, and A L t ) is the accrued interest on the reference obligation at time t as a percent of its face value. 6

7 A General Analysis Assuming Defaults at Discrete Times We now present a general analysis that can be used in conjunction with alternative N bonds assumptions about the claim amount. We assume that we have chosen a set of that are either issued by the reference entity or issued by another corporation that is 2 considered to have the same risk of default as the reference entity. We assume that defaults can happen on any of the bond maturity dates. Later we generalize the analysis i th bond is t to allow defaults to occur on any date. Suppose that the maturity of the i with

8 the th bond is j ( t ). If there is a default at time t , the bondholder makes a recovery at F j ( t ) on a claim of C ( t ). It follows that rate R j j α = v ( t (2) )[ F )] ( t t ) − R ( ( t C ) i j i i j i ij j p being incurred. The total present value of the of the loss α There is a probability, ij i losses on the j th bond is, therefore, given by: j ∑ G = α B (3) p − ij j i j =1 i p ’s to be determined inductively: This equation allows the ∑ 1 − j G B p − α − j j i ij i =1 p = (4) j α jj Recovery Rate Assumption These results have been produced on the assumption that interest rates are constant, recovery rates are known, and claim amounts are known. In what follows we will consider two assumptions about the claim amount. The first is that it equals the no-default value of the bond at the time of the default; the second is that it equals the face value plus accrued interest at the time of the default. It can be shown that, for either of these two assumptions, if a) default events, b) Treasury interest rates, and c) recovery rates are mutually independent, equations (2) and (3) are still true for stochastic interest rates, uncertain recovery rates, and uncertain default probabilities providing the recovery rate is set equal to its expected value in a risk-neutral world. It is probably reasonable to assume that there is no systematic risk in recovery rates so that expected recovery rates observed in the real world are also expected recovery rates in the risk-neutral world. This allows the expected recovery rate to be estimated from 3 historical data. Table 1 shows some estimates produced recently by Moody’s. As might 3 Moody’s calculates the recovery rate as the market value of the bond one month after default as a percent of its par value. The relevant recovery rate for our analysis is slightly lower. It is the market value of the bond one month after default as a percent of its par value plus accrued interest at the time of default. 8

9 be expected, the mean recovery rate is heavily dependent on the seniority of the bond. As mentioned earlier, the N bonds used in the analysis are issued either by the ref- erence entity or by another company that is considered to have the same risk of default should be the same for all bonds. The p as the reference entity. This means that the i recovery rates can in theory vary according to the bond and the default time. We will assume, for ease of exposition, that all the bonds have the same seniority in the event of default by the reference obligation and that the expected recovery rate is independent of t ( t ) is then independent of both j and . We will denote R time. The expected value of j ˆ R . this expected value by Extension to Situation Where Defaults can Happen at any Time The analysis used to derive equation (4) assumes that default can take place only on ( t )∆ t as q bond maturity dates. We now extend it to allow defaults at any time. Define t and t +∆ t as seen at time zero. The variable the probability of default between times ( ) is not the same as the hazard (default intensity) rate. The hazard rate, h ( t ), is defined t q h ( t )∆ t is the probability of default between times t and t +∆ t as seen at time t so that ( t q ( t ) and h . The variables t ) are related assuming no default between time zero and time by ∫ t τ ( h dτ ) − 0 e ) h )= t ( q ( t Many credit risk models such as Duffie and Singleton (1997), Jarrow and Turnbull (1995), h ( t ). However, we find it convenient to and Lando (1998) are formulated in terms of q ( t ) rather than h ( t ). We will refer to q ( t )asthe default express our results in terms of . probability density 4 t for

10 a similar analysis to that used in deriving equation (4) gives: ∑ j − 1 G − β − B q ij j i j =1 i = (6) q j β jj can be estimated using standard procedures, such as Simpson’s rule, β The parameters ij for evaluating a definite integral. Claim Amounts and Value Additivity We now present a numerical example and investigate the impact of different assump- tions about the claim amount. As mentioned earlier, Jarrow and Turnbull (1995) and Hull and White (1995) assume that, in the event of a default, the bondholder claims the no- ). t ( t )= F ( C default value of the bond. This is an attractive assumption. It implies that j j ˆ β R , is then proportional to 1 − so that equation (6) can be used to The parameter, ij ˆ (1 − R ) directly from observable market variables. Furthermore, an analysis of q estimate i B is the sum equation (6) shows that, in this case, the value of the coupon-bearing bond j of the values of the underlying zero-coupon bonds. This property is referred to as value . It implies that it is theoretically correct to calculate zero curves for different additivity rating categories (AAA, AA, A, BBB, etc) from actively traded bonds and use them for pricing less actively traded bonds. ( t ) equals the face value of bond C As mentioned earlier, the best assumption is that j j plus accrued interest at time t . As pointed out by Jarrow and Turnbull (2000), value additivity does not apply when this assumption is made (except in the special case where the recovery rate is zero). This means that there is no zero-coupon yield curve that can be used to price corporate bonds exactly for a given set of assumptions about default probabilities and expected recovery rates. Table 2 provides hypothetical data on six bonds issued by a reference entity. The bonds have maturities ranging from one to ten years and the spreads of their yields over Treasury yields are typical of those for BBB-rated bonds. The coupons are assumed to be paid semiannually, the Treasury zero curve is assumed to be flat at 5% (semiannually 10

11 compounded), and the expected recovery rate is assumed to be 30%. Table 3 calculates the default probability densities for the two alternative assumptions about the claim amount. It can be seen that the two assumptions give similar results. This is usually the case. For the default probability densities to be markedly different, it would be necessary for the coupons on the bonds to be either very much greater or very much less than the risk-free 5 rate. Expected Recovery Rates and Bond Yields Default probability densities must be greater than zero. From equation (6) this means that 1 j − ∑ − (7) ≤ G β q B j ij j i =1 i It is also true that the cumulative probability of default must be less than 1. This means that j ∑ 1 q ≤ ( t ) − t i 1 i i − =1 i or 1 j − ∑ − − t t ( q ) ≤ 1 − ) t ( t q 1 j i j j i − − 1 i =1 i so that from equation (6) [ ] − j j − 1 1 ∑ ∑ β jj 1 − (8) − β t − ≥ G t ) q ( − q B i i 1 − ij j i j i − t t j j − 1 =1 =1 i i Equations (7) and (8) impose both an upper and lower bound on the yield on the bond maturing at time t once expected recovery rates and the yields on bonds maturing j at earlier times have been specified. In the example in Table 2, when the expected recovery rate is 30%, a 20-year bond with a coupon of 7% must have a yield between 6.50% and 9.57% when the claim amount equals the face value plus accrued interest. 5 In emerging markets the yields on bonds can be several hundred or even several thousand percent. However, the coupons on these bonds are usually relatively low. 11

12 In general, we can use equations (7) and (8) to test whether a set of bond yields are consistent with the recovery rate assumption. Inconsistencies indicate that either the expected recovery rate assumption is wrong or bonds are mispriced. 12

13 2. The Valuation We now move on to consider the valuation of a plain vanilla credit default swap with a $1 notional principal. We assume that default events, Treasury interest rates, and recovery rates are mutually independent. We also assume that the claim in the event of default is the face value plus accrued interest. Define : Life of credit default swap T q t ): Risk-neutral default probability density at time t ( ˆ : Expected recovery rate on the reference obligation in a risk-neutral world. As indicated R in the previous section, this is assumed to be independent of the time of the default ( t ). q and the same as the recovery rate on the bonds used to calculate u ): Present value of payments at the rate of $1 per year on payment dates between time ( t t zero and time ∗ ∗ e where t is the payment − t ( ): Present value of an accrual payment at time t equal to t t . t date immediately preceding time ( t ): Present value of $1 received at time v t w : Total payments per year made by credit default swap buyer s : Value of w that causes the credit default swap to have a value of zero π : The risk-neutral probability of no credit event during the life of the swap A ): Accrued interest on the reference obligation at time t as a percent of face value t ( The value of is one minus the probability that a credit event will occur by time T . π q ( t ): It can be calculated from ∫ T t dt ) ( q =1 π − 0 The payments last until a credit event or until time T , whichever is sooner. If a default )]. If there is no t ( t

14 value of the payments is, therefore: ∫ T w + q )[ u ( t )+ e ( t )] dt t wπu ( T ) ( 0 Given our assumption about the claim amount, equation (1) shows that the risk-neutral expected payoff from the CDS is ˆ ˆ ˆ ( t )] ( R =1 − R R − A − t [1 + A ) 1 The present value of the expected payoff from the CDS is ∫ T ˆ ˆ [1 R − A ( t ) q R ] − dt t ) v ( t ) ( 0 and the value of the credit default swap to the buyer is the present value of the expected payoff minus the present value of the payments made by the buyer or ∫ ∫ T T ˆ ˆ R − A ( t ) ) R ] q ( t ) v ( t ) dt − w [1 − T q ( t )[ u ( t )+ e ( t )] dt − πwu ( 0 0 s w that makes this expression zero: The CDS spread, , is the value of ∫ T ˆ ˆ [1 − ) R − A ( t dt t R ] q ( t ) v ( ) 0 = s (9) ∫ T q ( t )[ u ( t )+ e ( t )] dt + πu ( T ) 0 CDS spread is referred to as the credit default swap spread or s .Itis The variable the total of the payments per year, as a percent of the notional principal, for a newly issued credit default swap. Consider the data in Table 2 and suppose that the reference obligation ˆ 3. Equation R =0 . is a five-year bond paying a semiannual coupon of 10% per annum with s for a five-year credit default swap with semiannual payments to be (9) gives the value of 1.944%. This is an annualized spread because of the way w is defined. Payments equal to 0.972% of the CDS notional principal would be required every six months. Approximate No-Arbitrage Arguments There is an approximate no-arbitrage argument that can be used to understand the s . If an investor forms a portfolio of a T -year par yield bond issued by determinants of 14

15 the reference entity and the credit default swap, the investor has eliminated most of the risks associated with default on the bond. If is the yield to maturity on the bond, the y y . In the absence of arbitrage − s investor’s net annual return is (at least, approximately) -year Treasury par yield, which T opportunities this should be (again, approximately) the .If y − s x x , an arbitrageur will find it is significantly higher than we will denote by profitable to buy a T -year par yield bond issued by the reference entity, buy the credit T -year par yield Treasury bond. If y − s is significantly less default swap, and short a than , an arbitrageur will find it profitable to short a T -year par yield bond issued by the x T -year Treasury par yield bond. reference entity, sell the credit default swap, and buy a The argument just given suggests that should equal y − x . However, a close analysis s of it shows that the arbitrage is less than perfect. Define: ∗ : y − x s L : CDS notional principal ∗ ( -year par t ): The accrued interest as a percent of the face value at time t on a T A yield bond that is issued at time zero by the reference entity with the same underlying par payment dates as the swap. We will refer to this bond as the . yield corporate bond R : Realized recovery rate when a default happens We first consider the situation where the Treasury curve is flat and interest rates are ∗ for a credit default swap where the s constant. In this case the CDS spread is exactly ∗ payoff in the event of a credit event at time L [1 + A t is ( t )](1 − R ). To see this, consider the position of an investor who buys both the credit default swap and an amount of the ∗ . Using when the spread is s L underlying corporate par yield bond with a face value of ∗ x . The is the corporate par yield, y , minus the Treasury rate, s the notation above, investor receives exactly the same cash flows as those from a Treasury par yield bond until t T or a credit event, whichever is earlier. If a credit event occurs at time either time (0

16 from the CDS is ∗ ∗ t )](1 − R ) − L ) y − x )( t − t ( ( L [1 + A ∗ ∗ . Because A is the payment date immediately prior to time ( t )= t t where as before ∗ − t y ), this reduces to ( t ∗ ∗ x − [1 + t L ( t [1 + t LR ( − )] )] A ∗ LR A The corporate bond holding is worth [1 + ( t )] so that the net value of the holding is ∗ x ( t − t L [1 + )]. This is exactly what is required to buy a par yield Treasury bond with t . It follows that in all circumstances, the investor’s portfolio L at time a face value of ∗ exactly replicates the cash flows from the par yield Treasury bond showing that s must ∗ would give rise to an be the correct CDS spread. A spread greater than or less than s arbitrage opportunity. ∗ ( )](1 t − R )asan idealized A We will refer to a CDS that provides a payoff of [1 + ∗ . Our analysis shows that the spread on such a CDS is exactly s . credit default swap − R − A ( t ) R rather than In practice, the payoff from a credit default swap is usually 1 ∗ ∗ )](1 − R ). This leads to t s overestimating the true spread, s . ( A [1 + Continuing for a moment with the assumption that the Treasury curve is flat and interest rates are constant, we can correct for the difference between the payoff on the idealized CDS and the actual CDS. An analysis similar to that leading up to equation (9) shows that the spread for an idealized credit default swap is given by ∫ T ∗ ˆ (1 ) − R dt ) ( t A [1 + ( t )] q ( t ) v ∗ 0 s = ∫ T ( t )[ u ( t )+ e ( t )] q + πu ( T ) dt 0 An approximation to this is ∫ T ∗ ˆ (1 R )(1 + a − v ) dt ) q ( t ) t ( ∗ 0 s = (10) ∫ T q ( t )[ u ( t )+ e ( t )] dt + πu ( T ) 0 ∗ ∗ is the average value of A . Similarly, from equation (9), an ( t ) for 0 ≤ t ≤ T a where approximation to the actual CDS spread is ∫ T ˆ ˆ (1 − R ) R − a ( dt t q ) t ) v ( 0 s = (11) ∫ T q ( t )[ u ( t )+ e ( t )] dt + πu ( T ) 0 16

17 ≤ where ( t ) for 0 ≤ t A T . is the average value of a From equations (9) and (10) ∗ ˆ ˆ R (1 R − a − ) s (12) = s ∗ ˆ )(1 + (1 − ) a R As an illustration of equation (12), consider the data in Table 2 and assume, as before, that the coupon on the reference obligation is 10%. (We will refer to this as Case A; see Table 4.) The five-year par yield for bonds issued by the reference entity is 7%. The five-year Treasury par yield is 5%. It follows that, for a five-year credit default swap with ∗ is 2.00%. The coupon paid every six months on a par yield bond s semiannual payments, ∗ a issued by the reference entity is 3.5 per 100 of principal so that =0 . a =0 . 025 0175. Also ˆ . =0 945%. This is very close to the 1.944% 3 so that equation (12) gives s R . =1 and estimate reported earlier from using equation (9). Equation (12) assumes a flat Treasury yield curve and constant interest rates. Stochas- tic interest rates make the no-arbitrage argument for the idealized CDS less than perfect, but do not affect valuations given our assumption that interest rates, default probabilities, and recovery rates are independent. However, the no-arbitrage argument for the idealized CDS swap requires a flat yield curve so that a par yield Treasury bond is always worth its face value plus accrued interest at the time of a default. An upward sloping yield curve will lead to the par yield Treasury bond being worth less than the face value plus accrued ∗ underestimates the spread for the idealized CDS. Simi- s interest on average. As a result ∗ s larly a downward sloping yield curve leads to overestimating the spread on the idealized CDS. As pointed out by Duffie (1999), we can deal with non-flat Treasury curves by con- sidering par yield floating-rate bonds rather than par yield fixed-rate bonds. Define a Treasury par floater as a floating-rate bond where the interest rate is reset on each pay- ment date of the credit default swap, and a par floater issued by the reference entity as a similar floating-rate bond that promises a prespecified spread above the Treasury par 17

18 floater for the life of the credit default swap. If the payoff from the credit default swap is ∗ ∗ t )](1 − ) is here defined as the accrual on the par floater issued by ) where A ( ( t R [1 + A the reference entity, the arbitrage arguments are watertight and the CDS spread should exactly equal the spread of the reference entity floater over the Treasury floater. In practice we rarely get the opportunity to observe the spreads on corporate par yield floaters. Credit default swaps must be evaluated from the yields on fixed rate bonds issued by the reference entity. The difference between the spread on par yield floaters and par yield fixed rate instruments is very small for flat term structures, but noticeable for non-flat term structures. As an extreme test of the effect of a non-flat term structure we changed the flat Treasury curve in Case A to a Treasury curve where the 1-, 2-, 3-, 4-, and 5-year par yields were 1%, 2%, 3%, 4%, and 5%, respectively. (We will refer to this as Case B; see Table 4.) Everything else, including the spreads between par yields on Treasuries and yields on bonds issued by the reference entity was maintained as in Case A. As a result, the five-year par yield for bonds issued by the reference was still 7% and ∗ was still 2.00%. However, the value of s given by equation (9) increased from 1.944% s to 2.071%. We also rarely get the chance to observe corporate bonds that are selling for exactly their par value. Assuming that the yield on a non-par-yield bond is the same as the yield on a par yield bond introduces some error. We tested this by changing the coupons on all bonds used to calculate default probabilities in Case A from 7% to 4% while keeping everything else (including the yield on the bonds) the same as in Case A. (We will refer s increased from 1.944 to 1.990. This change to this as Case C; see Table 4.) The value of 6 For less results entirely from the correct five year par yield being 7.048% rather than 7%. creditworthy reference entities, the error from basing calculations on non-par-yield bonds 6 s when the correct par yield is used Equation (12) provides an accurate estimate of ∗ ∗ ˆ =2 3. 048, a . =0 . 01762, a =0 . 025 and . R =0 s so that 18

19 can be much greater. Suppose Case A is changed so that the recovery rate is zero and the 1-, 2-, 3-, 4-, and 5-year yields on bonds issued by the reference entity are 10%, 20% 30%, 40% and 50%, respectively. (We will refer to this as Case D; see Table 4.) Assuming that the par ∗ = 45, ( s s yield is 50% and using equation (12) leads to an estimate of 40% for the value of ∗ ˆ . a 125, =0 . 025, and =0 R = 0). The correct value of s given by equation (9) is 29.98%. a This difference largely results from the correct par yield being about 38% rather than 50%. Impact of Expected Recovery Rate on Pricing The one parameter necessary for valuing a credit default swap that cannot be observed directly in the market is the expected recovery rate. We assume that the same recovery rate is used for estimating the default probability densities and for calculating the payoff. As it happens there is an offset. As the expected recovery rate increases, estimates of the probability of default increase and payoffs decrease. The overall impact of the recovery rate assumption on the value of a credit default swap is generally fairly small when the expected recovery rate is in the 0% to 50% range. This is illustrated in Figure 1, which shows the dependence of the five-year CDS spread 7 In Case A the yield on the expected recovery rate in Cases A, B, and C considered above. ∗ , is always s curve is flat and the par yield spread is 2%. The spread for an idealized CDS, exactly 2%. The actual CDS spread is less than 2% and is a decreasing function of the expected recovery rate. When we move from Case A to the upward sloping yield curve in Case B, the CDS spread increases. The higher the expected recovery rate, the greater the impact of an upward sloping yield curve on the CDS spread. In Case C the coupon on the five year bond is less than the five-year par yield. This leads to the par yield being an increasing function of the expected recovery rate. As a result the CDS spread is also an increasing function of the expected recovery rate. 7 We did not consider Case D because equations (5) and (8) imply that the expected recovery rate must be very low. 19

20 Binary Credit Default Swaps A binary credit default swap is structured similarly to a regular credit default swap except that the payoff is a fixed dollar amount. A similar analysis to that given earlier shows that the value a binary CDS spread that provides a payoff of $1 in the event of a default is ∫ T q ( t dt v ( t ) ) 0 ∫ T ) q ( t )[ u ( t )+ e ( t )] dt + πu ( T 0 This is quite heavily dependent on the expected recovery rate, as illustrated by Figure 2. The Independence Assumptions The valuation approaches we have presented are based on the assumption that inter- est rates, default probabilities, and recovery rates are independent. These assumptions are unlikely to be perfectly true in practice. For example, it can be argued that high interest rates cause companies to experience financial difficulties and, as a result, default probabil- ities increase. Such a positive relation between interest rates and default probabilities has two effects. First, high default probabilities tend to be associated with high discount rates for the payoffs. This reduces the CDS spread. Second high default probabilities tend to be associated with relatively low market values for bonds issued by the reference entity. This increases the CDS spread (because it increases the value of the buyer’s right to sell the reference bond for its face value). It is reassuring that these effects act in opposite directions so there is a partial offset. Note that the relevant correlation for the first effect and the average short term interest rates between time t is between default rates at time t ; the relevant correlation for the second effect is between interest rates at zero and time . As far as the second effect is concerned, the t and medium to long rates at time t time correlation is less than might be supposed because there are often significant time lags 8 between the occurrence of high interest rates and the resultant defaults. 8 For example, high interest rates in the early 1980s caused problems in many sectors of the U.S. economy, but many of the resulting defaults occurred a few years later—when rates were much lower. 20

21 Moody’s Investor’s Service (2000) provides statistics which suggest that the correla- tions are small and provides a reasonable comfort level for the independence assumptions. Default rates are only weakly correlated with macroeconomic variables. For example, the correlation between the US Industrial Production Index and the All Corporate Default Rate is reported to be -0.14. Moody’s does provide some evidence that recovery rates are positively correlated with general economic conditions. This suggests that recovery rates may be negatively related to default probabilities—a phenomenon that increases CDS spreads. However, again we can reasonably hypothesize that the effect is small. 21

22 3. Application to Real Data Up to now our illustrations of the methodology have been somewhat idealized. We now apply the approach to a real-world data. We consider the valuation of credit default swaps on Ashland Inc. at the close of trading on July 13, 2000. Ashland is a Fortune 500 company based in Kentucky with interests in chemicals, oil, car products, petroleum refining and retailing, and coal. The company had about 80 unsecured bonds outstanding on July 13, 2000. We did not have data on the volume of trading for the bonds and chose to base our analysis on quotes for a sample of eight of the bonds that had a wide range of maturities and relatively high issue sizes. Quotes for these bonds at the close of trading on July 13, 2000 are shown in Table 5. All bonds in our sample are all rated BBB by S&P and Baa2 by Moody’s. They contain no embedded options apart from a poison put. Quotes for the benchmark Treasury bonds and Treasury bills at the close of trading on July 13, 2000 are shown in Table 6. We used a bootstrap procedure to calculate a Treasury zero curve from the data in Table 6. We then estimated the default probabilities shown in Table 7 using the approach in Section 1. This in turn enabled us to use the approach in Section 2 to calculate CDS spreads for instruments with semiannual payments and a number of different maturities. The reference obligation was assumed to have a coupon of 8% per annum paid semiannually. From Table 1 we estimated the expected recovery rate to be 48.84%. Table 8 shows the CDS spreads together with (a) The spread between Treasury yields and Ashland yields; and (b) The spread between Treasury yields and Ashland yields after the adjustment in equa- tion (12) is applied To calculate the spread in (a), we interpolated between quoted Treasury yields to calculate Treasury yields for bonds with the same maturity as the Ashland bonds. This enabled us to calculate a yield spread for each Ashland bond. We then interpolated between these yield spreads to calculate yield spreads for the CDS maturities. 22

23 ∗ To make the adjustment in (b), ( t ) was estimated by bootstrapping a zero curve A for Ashland and using it to calculate par yields. This involved using the bootstrap method to calculate a zero curve for Ashland and using this to calculate the par yields. There is a very small approximation here because, as mentioned in Section 1, value additivity does not apply to corporate bonds. Table 8 shows that the approach in equation (12) works reasonably well. It provides a significant improvement over the naive approach of setting the CDS spread equal to the interpolated credit spread. The CDS spread estimate given by equation (12) is higher than the true CDS spread for the 20-year maturity because the long end of the Treasury yield curve is downward sloping. 23

24 4. Conclusions The valuation of a credit default swap is a two step procedure. First bonds issued by the reference entity (or a company with the same risk of default as the reference entity) must be used to estimate the risk-neutral probability of the reference entity defaulting at different future times. The present value of the expected payments made by the buyer of the swap and the present value of the expected payoff must then be calculated. The valuation requires estimates of the amount claimed by bondholders in the event of a default and the expected recovery rate. The most realistic assumption about the amount claimed in the event of a default is that it equals the face value of the bond plus accrued interest. The expected recovery rate must be estimated from empirical data. The valuation of a vanilla CDS is relatively insensitive to the expected recovery rate, but this is not so for a binary CDS. T -year credit default swap spread is the yield on a T -year A simple estimate of the T -year Treasury par yield. A small adjust- bond issued by the reference entity minus the ment to the estimate, equation (12), should be made to reflect the way the payoffs on credit default swaps are calculated. The estimate is then reasonably accurate in many circumstances. However, there are errors when the Treasury zero curve is significantly non-flat and when rates are very high. 24

25 REFERENCES Duffie, D. “Credit Swap Valuation,” , (January/February Financial Analysts Journal 1999), 73–87. Duffie, D. and K. J. Singleton, “An Econometric Model of the Term Structure of Interest-Rate Swap Yields,” Journal of Finance 52, 1 (September 1997), 1287–1321. Hull, J. C. and A. White, “The Impact of Default Risk on Options and Other Deriva- tive Securities,” , 19,2 (1995), 299–322. Journal of Banking and Finance Hull, J. C. and A. White, “Valuing Credit Default Swaps II: Modeling Default Cor- relations,” Working Paper in progress, University of Toronto, 2000. Jarrow, R. A. and S. Turnbull, “Pricing Options on Derivative Securities Subject to Credit Risk,” Journal of Finance , 50, 1, (1995), 53–85. Jarrow, R. A. and S. Turnbull, “The Intersection of Market and Credit Risk,” Journal of Banking and Finance , 24 (2000), 271–299. J.P. Morgan, The J.P. Morgan Guide to Credit Derivatives , Risk Publications, 1999. Lando, D., “On Cox Processes and Credit Risky Securities”, Review of Derivatives Research , 2 (1998), 99–120. Moody’s Investor’s Service, “Historical Default Rates of Corporate Bond Issuers, 1920–99,” (January 2000). 25

26 Table 1 Recovery Rates on Corporate Bonds from Moody’s Investor’s Service (2000) Class Mean (%) Standard Deviation (%) Senior Secured 52.31 25.15 Senior Unsecured 48.84 25.01 Senior Subordinated 39.46 24.59 Subordinated 33.17 20.78 Junior Subordinated 19.69 13.85 26

27 Table 2 ∗ Hypothetical Example of Bonds Issued by Reference Entity Coupon Bond Life Bond Yield (%) (Spread Over Treasury Par Yield in bps) (years) 1 160 7.0 2 7.0 170 3 7.0 180 4 190 7.0 5 7.0 200 10 7.0 220 ∗ Bond coupon is paid semiannually and bond yield is expressed with semiannual com- pounding. All Treasury rates are assumed to be 5% per annum with semiannual com- pounding. Expected recovery rate = 30%. 27

28 Table 3 Implied Probabilities of Default for Data in Table 2 Default Probability Density Time Claim = No-default Value Claim = Face Value + Accr. Int. (years) 0–1 0.0219 0.0220 1–2 0.0245 0.0242 2–3 0.0269 0.0264 3–4 0.0292 0.0285 4–5 0.0315 0.0305 5–10 0.0295 0.0279 28

29 Table 4 ∗ Cases Considered CDS Spread CDS Spread Data Used equation (9) equation (12) Case A All Treasury rates are 5%; spreads 1.944 1.945 and coupons on corporate bonds are as in Table 2; recovery rate is 30% 2.071 1.945 Case B 1-, 2-, 3-, 4-, and 5-year Treasury par yields are 1%, 2%, 3%, 4%, and 5% respectively; spreads on corporate bonds are as in Table 2; recovery rate is 30% Case C All Treasury rates are 5%; spreads 1.990 1.945 on corporate bonds are as in Table 2; coupons on corporate bonds are 4%; recovery rate is 30% Case D All Treasury rates are 5%; coupons 29.98 40.00 on corporate bonds are as in Table 2; yields on 1-, 2-, 3-, 4-, and 5-year corporate bonds are 10%, 20%, 30%, 40%, and 50%; recovery rate is 0% ∗ Coupons are paid semiannually and all rates and yields are expressed with semiannual compounding. 29

30 Table 5 Quotes for Unsecured Bonds Issued By Ashland Inc. at Close of Trading on July 13, 2000 Maturity Quoted Quoted Coupon Date % per annum Price Yield 9.48 Dec 15, 2000 7.715 100.672 Mar 1, 2001 9.30 100.689 8.150 Jan 27, 2003 8.40 100.234 8.295 Jul 21, 2004 7.91 98.899 8.237 Nov 14, 2006 93.066 8.332 6.90 Dec 27, 2011 8.88 103.067 8.455 Apr 1, 2015 8.38 98.433 8.569 Feb 21, 2025 8.63 100.105 8.619 Source: Bridge–Telerate Service in the Financial Research and Trading Lab, Joseph L. Rotman School of Management, University of Toronto. 30

31 Table 6 Quotes for Benchmark Government Bills and Bonds at Close of Trading on July 13, 2000 Maturity Quoted Quoted Coupon % per annum Price Yield Date Bill 5.990 6.171 Oct 12, 2000 Jan 11, 2001 Bill 5.990 6.267 May 31, 2001 Bill 5.740 6.055 Jun 30, 2002 100.141 6.296 6.375 May 15, 2005 6.750 102.516 6.140 Feb 15, 2010 6.500 103.563 6.005 May 15, 2030 6.250 106.094 5.817 Source: Bridge–Telerate Service in the Financial Research and Trading Lab, Joseph L. Rotman School of Management, University of Toronto. 31

32 Table 7 Cumulative Risk-Neutral Default Probabilities for Ashland Inc. Date Cumulative Default Probability Dec 15, 2000 0.0124 Mar 1, 2001 0.0231 Jan 27, 2003 0.0929 Jul 21, 2004 0.1455 Nov 14, 2006 0.2472 Dec 27, 2011 0.4183 Apr 1, 2015 0.5563 Feb 21, 2025 0.7642 32

33 Table 8 CDS Spreads for Ashland and Estimates Provided by the Yield Spread Before and After Applying the Adjustment in Equation (12) Maturity Adjusted Yield Spread CDS Spread Yield Spread Equation (12) Equation (9) (years) 199 191 189 1 2 202 194 193 3 204 196 196 4 197 198 205 5 213 205 209 10 240 231 227 15 262 251 251 20 269 258 253 33

34 2.5 Case A 2 Case B Case C 1.5 Vanilla CDS Spread (%) 00.20.40.60.8 Ex pe c t e d Re c ove r y Ra te Figure 1 Dependence of Five-Year Credit Default Swap Spread on the Expected Recovery Rate When Reference Liability is a Five-Year 10% Coupon Bond Case A: Data as in Table 2 Case B: Data as in Table 2 except that one-, two-, three-, four-, and five-year Treasury par yields are 1%, 2%, 3%, 4%, and 5%, respectively Case C: Data as in Table 2 except that the coupons on all bonds are 4% instead of 7% 34

35 14 12 10 Case A 8 Case B 6 Case C 4 2 0 Binary CDS Spread (%) 0.8 0.6 0.4 0 0.2 y Rate Expected Recover Figure 2 Dependence of Five-Year Binary Credit Default Swap Spread on the Expected Recovery Rate Case A: Data as in Table 2 Case B: Data as in Table 2 except that one-, two-, three-, four-, and five-year Treasury par yields are 1%, 2%, 3%, 4%, 5%, respectively Case C: Data as in Table 2 except that the coupons on all bonds are 4% instead of 7% 35