See J. C. Hull and A. White, “Valuing Credit Default Swaps 1: No Counterparty Default Risk,” Journal ofDerivatives, 8, no. | (Falf 2000), 29-40.642 CHAPTER 27
 
times t;_, and t;. We use the notation introduced earlier with the modification that A,(t) is the
acerned interest on the ith reference bond at time ¢.
From equation (27.2),_ Year He fit [L — R- ARO at© Der i Ak (WD + edt + ult ll — Dear ele — GV
 
The g; can be calculated inductively from this equation. Define 5, = t, — t_, and
a = I (1 — Ryo(t) dt -
bk
By = I A,()Re(t) de
ta4
n= | tue +etmarey
It follows that
 
a= siu(ts) + Vihar Uelsive — 8H), = Oe + Beil (27.4)
‘ at; — Bi y — Sry + S;U(t;)3;
Example 27.4 Suppose that the 3-, 5-, 7-, and 10-year swap spreads for a particular referenceentity are 125, 155, 173, and 193 basis points, respectively. Payments are made semiannually and thereference bonds are all 7% bonds that pay coupons semiannually. The LIBOR curve is flat at 5%with continuous compounding and the expected recovery rate is 48%. Equation (27.4) shows that thedefault probability is 2.33% per year for the first three years, 3.63% per year for years 4 and 5, 3.73%per year for years 6 and 7, and 3.79% per year for years 8, 9, and 10.
Recovery Rate Estimates
The only variable necessary for valuing a credit default swap that cannot be observed directly inthe market is the expected recovery rate. Fortunately the pricing of a plain vanilla credit defaultswap depends on the recovery rate to only a smail extent. This is because the expected recovery rateaffects credit default swap prices in two ways. It affects the estimates of risk-neutral defaultprobabilities and the estimates of the payoff that will be made in the event of a default. Thesetwo effects largely offset each other.’ In Example 27.2, we calculated a five-year CDS spread basedon the data in Table 26.6 as 1.944% of the principal or 194.4 basis points. This was based on anexpected recovery rate of 30%. If the expected recovery rate is increased to 50%, the CDS spreadchanges to 191.6; if it decreases to 10%, the CDS spread changes to 196.0.
Binary Credit Default Swaps
Nonstandard credit default swaps can be quite sensitive to the expected recovery rate estimate.Consider a binary credit default swap. This is structured similarly to a regular credit default swap
> This is true both when the risk-neutral default probabilities are estimated from a set of bond prices, as they are inSection 26.1, and when they are estimated from a set of credit default swap spreads as in equation (27.4).
+ Note that the probabilities of default in Table 26.6 change when these changes are made.Credit Derivatives 643
except that the payoff is a fixed dollar amount. In this case the expected recovery rate affects theprobability of default but not the payoff. As a result the credit default spread is quite sensitive to therecovery rate. For example, a binary credit default swap spread when the expected recovery rate is50% is typically about 80% higher than when the expected recovery rate is 10% (see Problem 27.5).
Basket Credit Default Swaps
In a basket credit default swap there are a number of reference entities. An add-up basket creditdefault swap provides a payoff when any of the reference entities default. It is equivalent to aportfolio of credit default swaps, one on each reference entity. A first-to-default basket creditdefault swap provides a payoff only when the first reference entity defaults. After that, there are nofurther payments on the swap and it ceases to exist.
First-to-default swaps can be valued using Monte Carlo simulation. On each trial each referenceentity is simulated to determine when, if ever, it defaults. We calculate (a) the present value of thepayoff (if any) and (b) the present value of payments until the time of the first default or the end ofthe contract (whichever is earlier) at the rate of $1 per year. The swap spread is the average value ofthe calculations in (a) divided by the average value of the calculations in (b). First-to-default swapsare sensitive to the default correlation between reference entities. The higher the correlation, thelower the value. A conservative assumption for the seller of the swap is that all correlations arezero. Nonzero default correlations can be incorporated into the Monte Carlo simulation using theGaussian copula model explained in Section 26.8.
Seller Default Risk
The analysis we have presented so far assumes that there is no chance that the seller of a creditdefault swap will default. In this section we outline how the impact of seller default risk on CDSspreads can be estimated.
To understand the nature of seller default risk, consider the position of the buyer immediatelyafter the seller defaults. (We assume that the reference entity has not previously defaulted.) Tomaintain the default protection, the buyer must immediately buy a new CDS from a new seller.The life of the new CDS will be T — ty, where T is the maturity of the original CDS and fp is thetime at which the default occurs. The buyer will take a loss if the amount paid per annum forprotection under the new contract is greater than that paid under the old contract. This is likely tobe the case if the creditworthiness of the reference entity has declined significantly. Seller defaultrisk therefore depends on:
1. The extent to which the reference entity’s default probability is expected to increase with time2. The default correlation between the reference entity and the seller
The impact of sefler default risk can be quantified using Monte Carlo simulation. Define Y as thepresent value of the payoff from the CDS and C as the present value of payments made at the rateof $1 per year until either the end of the life of the CDS or until default. On each run of thesimulation, we sample the times (if any) of defaults for the reference entity and the counterparty.°This enables sample values for Y and C to be obtained. If neither the reference entity nor the sellerdefaults, then ¥ = 0 and C is the present value of the $1 annuity for the whole life of the CDS. Ifthe reference entity defaults first, then Y is the payoff from the CDS and C is the present value of
° The Gaussian copula model for times to default described in Section 26.7 can be used.644 CHAPTER 27
the $1 annuity until the reference entity’s default. If the seller defaults first, then Y = 0 and C is thepresent value of a $1 annuity uniil the selier defauits.° The credit default swap spread is the averagevalue of Y divided by the average value of C.
Hull and White provide some results from this type of analysis.’ They show that the impact ofdefault risk on the credit default swap spread is very low when the default correlation between theseller and the reference entity is zero. As the correlation rises, the possibility of a seller defaultbecomes more significant. Clearly it makes sense to buy a CDS from a company whose defaultcorrelation with the reference entity is as low as possible.
27.2. TOTAL RETURN SWAPS
 
 
A total return swap is an agreement to exchange the total return on a bond or other reference assetfor LIBOR plus a spread. The total return includes coupons, interest, and the gain or loss on theasset over the life of the swap.
Anexample of a total return swap is a five-year agreement with a notional principal of $100 millionto exchange the total return on a 5% coupon bond for LIBOR pius 25 basis points. This is illustratedin Figure 27.2. On coupon payment dates, the payer pays the coupons earned on an investment of$100 million in the bond. The receiver pays interest at a rate of LIBOR plus 25 basis points on aprincipal of $100 million. (LIBOR is set on one coupon date and paid on the next as in a vanillainterest rate swap.) At the end of the life of the swap, there is a payment reflecting the change in valueof the bond. For example, if the bond increases in value by 10% over the life of the swap, the payer isrequired to pay $10 million (= 10% of $100 million) at the end of the five years. Similarly, if the bonddecreases in value by 15%, the receiver is required to pay $15 million at the end of the five years. Ifthere is a default on the bond, the swap is usually terminated and the receiver makes a final paymentequal to the excess of $100 million over the market value of the bond.
If we add the notional principal to both sides at the end of the fife of the swap, we cancharacterize the total return swap as follows. The payer pays the cash flows on an investment of$100 in the 5% corporate bond. The receiver pays the cash flows on a $100 million bond payingLIBOR plus 25 basis points. If the payer owns the bond, the total return swap allows it to pass thecredit risk on the bond to the receiver. If it does not own the bond, the total return swap allows itto take a short position in the bond.
Total return swaps are usually used as a financing tools. The most likely scenario leading to theswap in Figure 27.2 is as follows. The receiver wants financing to invest $100 million in thereference bond. It approaches the payer (which is likely to be a financial institution) and agrees tothe swap. The payer then invests $100 million in the bond. This leaves the receiver in the same
Total Total return on bond - Totalreturn - return
payer LIBOR + 25 basis points receiver
 
 
 
Figure 27.2 Total return swap
© This is somewhat conservative as the buyer of a CDS may in some circumstances be able to make a claim againstthe assets of the seller in the event of a default by the seller.
7 See J. C. Hull and A, White, “Valuing Credit Default Swaps I: Modeling Default Correlations,” Journal ofDerivatives, 8, no. 3 (Spring 2001), 12-22.Credit Derivatives 645
position as it would have been if it had borrowed money at LIBOR plus 25 basis points to buy thebond. The payer retains ownership of the bond for the life of the swap and has much less exposureto the risk of the receiver defaulting than it Would have if it had lent money to the receiver tofinance the purchase of the bond. Total return swaps are similar to repos (see Section 5.1) in thatthey are structured to minimize credit risk when money is borrowed.
If we assume that there is no risk of a default by either the recciver or the payer, the valuc of theswap to the receiver at any time should be the value of the investment in the reference bond less thevalue of the $100 million LIBOR bond. Similarly the value of the swap to the payer should be thevalue of the LIBOR bond less the value of the $100 million investment in the reference bond.When it is first entered into the value of the swap should be zero. The value of the investment inthe reference bond is $100 million. The value of the LIBOR bond should therefore also be$100 million. This suggests that the spread over LIBOR should be zero rather than 25 basis points.
In vractice, the naver is likely to require a spread above LIBOR as comnensation for bearins thean practice, tne payer is ukery tO require & spread acove LasUus as compensation for oearing tne
risk that the receiver will default. The payer will lose money if the receiver defaults at a time whenthe reference bond’s price has declined. The spread therefore depends on the credit quality of thereceiver, the credit quality of the bond issuer, and the default correlation between the two.
There are a number of variations on the standard deal we have described. Sometimes, instead ofthere being a cash payment for the change in value of the bond, there is physical settlement wherethe payer exchanges the underlying asset for the notional principal at the end of the life of theswap. Sometimes the change-in-value payments are made periodically rather than all at the end.The swap then has similarities to an equity swap (see Section 25.5).
27.3. CREDIT SPREAD OPTIONS
 
Credit spread options are options where the payoff depends on either a particular credit spread or theprice of a credit-sensitive asset. Typically the options are structured so that they cease to exist if theunderlying asset defaults. If a trader wants protection against both an increase in a spread and adefault, then both a credit spread option and another instrument, such as a credit default swap, arerequired.
One type of credit spread option is defined so that it has a payoff of
Dmax(K — Sr, 0)
or_ Dmax(S; — K, 0)where Sv is a particular credit spread at option maturity, K is the strike spread, and — is a durationused to translate the spread into a price. This can be valued using Black’s model (with appropriateconvexity adjustments) by assuming that the future spread conditional on no default is lognormal.The usual Black’s model formula must be multiplied by the probability of no default during the lifeof the option.
Another type of credit spread option is a European call or put option on a credit-sensitive assetject aca flnatino-rate nate In this case the navoff is either
max(S; — K, 0)ormax(K — S7, 0)646 CHAPTER 27
where S, is the price of the asset at maturity and K is the strike price. This can be valued usingBlack’s model by assuming that, conditional on no default, the credit-sensitive asset price islognormal at option maturity. As in the previous case, the usual Black’s model formula must bemultiplied by the probability of no default during the life of the option.
Credit spread options are sometimes embedded in other products. Examples of this are:
1. A guarantee that the spread over LIBOR in a floating-rate loan will not rise above a certainlevel
. A prepayable floating-rate loan where the spread over LIBOR is fixed
. The right to enter into an asset swap
. The right to terminate an asset swap
. The right to enter into a credit default swap
. The right to terminate a credit default swap
Dm Wt
27.4 COLLATERALIZED DEBT OBLIGATIONS
 
A collateralized debt obligation (CDO) is a way of packaging credit risk in much the same way as acollateralized mortgage obligation (see Section 24.4) is a way of packaging prepayment risk. A typicalstructure is shown in Figure 27.3. This involves creating four classes of securities, known as tranches,from a portfolio of corporate bonds or bank loans. The first tranche has 5% of the principal of theportfolio and absorbs the first 5% of the default losses. The second tranche has 10% of the principaland absorbs the next 10% of the default losses. The third tranche has 10% of the principal andabsorbs the next 10% of the default losses. The fourth tranche has 75% of the bond principal andabsorbs the residual default losses.
Tranche 1Ist 5% of lossYield = 35%
Tranche 22nd 10% of lossYield = 15%
                  
ItIY Tranche 33rd 10% of loss
Bond n Yield = 7.5%
Tranche 4Residual loss
Yield = 6%
Average yield8.5%
Figure 27.3 Collateralized debt obligationCredit Derivatives 647
Tranche 4 will usually be rated Aaa by Moody’s and AAA by S&P, because there is very littlechance of any default losses on this tranche. Default losses on the bond portfolio would have toexceed 25% of the principal before investors in tranche 4 are affected. Tranche 3 has more defaultrisk than tranche 4, but less default risk than the original portfolio. Default risks on the originalportfolio must exceed 15% of principal for the holders of tranche 3 to be affected. Tranche 2 islikely to be more risky than the original portfolio. When default losses are between 5% and 15% ofthe original portfolio, they are absorbed by the holders of Tranche 2. Tranche | is sometimesreferred to as “toxic waste’. It is subject to significant default risk. A default loss of 2.5% on theoriginal portfolio translates to a default loss of 50% on tranche 1. A default loss of 5% or more onthe original portfolio leads to a default loss of 100% on tranche 1. The yields in Figure 27.3, like allbond yields, are promised yields not expected yields. They do not take account of expected defaults.
The creator of the CDO normally retains tranche 1 and sells the remaining tranches in themarket.2 A CDO provides a way of creating high-quality debt from average-quality (or even low-quality) debt. The risk to the purchaser of tranches 2, 3, or 4 depends on the default correlationbetween the issuers of the debt instruments in the portfolio. The lower the correlation, the morehighly tranches 2, 3, and 4 will be rated. Default correlation models such as the Gaussian copulamodel discussed in Section 26.8 are often used to analyze CDOs.
275 Aaia an
 
 
Suppose that you use Black Scholes (with an appropriate volatility smile) to price a two-yearEuropean option you are planning to buy and conclude that the option is worth $6. You thendiscover that the counterparty selling you the option is A-rated. How do you adjust the $6 price totake account of the possibility of the counterparty defaulting?
Adjusting the pricing of a derivative such as the one we have just mentioned can be verycomplicated. If there are netting agreements, the adjustment should depend on the other contractsoutstanding with the counterparty. The adjustment should also depend on any collateralizationagreements and downgrade triggers that have been negotiated (see Section 26.6 for a discussionof this).
In this section we show how the price of a derivative should be adjusted if it is treated as a stand-alone transaction with no netting, downgrade triggers, and collateralization. This is sometimesused as a guide to the bid~offer spreads that should be negotiated. We assume that the followingtwo sets of variables are mutually independent:
1. The variables affecting the value of the derivative value in a no-default world.
2. The variables affecting the occurrence of defaults by the counterparty and the percentagerecovery made in the event of a default
We also assume that the claim amount in the event of default is the no-default value of thederivative and define:
f(t): Value of the derivative at time t taking account of the possibility of a default by the
counterparty
ounterpa:
f°): Value of a similar default-free contract at time ¢
5 Tranche | is usually difficult to sell.648 CHAPTER 27
 
Contracts That Are Assets
We start by considering a derivative that provides a payoff at time T and is always an asset to usand a liability to the counterparty. (An example is the two-year option considered earlier.) It canbe shown that
FO) = Fe ROOT (27.5)
In this equation, y(T) is the zero-coupon yield on a T-year zero-coupon bond, issued by thecounterparty and ranking equally with the contract under consideration, and y*(7) is the zero-coupon yield on a similar risk-free bond. To understand equation (27.5), compare the derivativewith a T-year zero-coupon bond issued by the counterparty. Whenever a default occurs, the loss asa percent of no-default value on both the bond and option is the same. Because the no-defaultvalue of the derivative and the bond are assumed to be independent of the probability of default,the percentage reduction in the price of the derivative because of default risk is the same as that onthe bond. The price of the bond is eO-MIT times its no-default price. The same must be trueof the option. Equation (27.5) follows.
Example 27.5 Consider a 2-year over-the-counter option with a default-free value of $3. Supposethat a 2-year bond issued by the option writer that ranks equal to the option in the event of a detaultyields 150 basis points over a similar risk-free bond. Default risk has the effect of reducing the optionprice to
3e-001S*2 9.94]or by about 3%.
Equation (27.5) shows that we can calculate f(0) by applying a discount rate of y— y* to f*(0).Because f*(0) is itself obtained by discounting the expected payoff in a risk-neutral world at y*, itfollows that f can be calculated by discounting the expected payoff in a risk-neutral world aty’ +(y—y*) = y. One interpretation of equation (27.5), therefore, is that we should use the“risky” discount rate, y, instead of the risk-free discount rate, y*, when discounting payoffs froma derivative contract.
Note that the risk-free interest rate enters into the valuation of a derivative in two ways. It isused to define the expected return from the underlying asset in a risk-neutral world, and it is usedto discount the expected payoff. We should change the risk-free rate to the risky rate fordiscounting purposes, but not when determining expected returns in a risk-neutral world. Forexample, when using a binomial tree, the growth rate of the underlying asset should be the risk-freerate, but we should use the risky rate for discounting as we work back through the tree.
Contracts That Can Be Assets or Liabilities
We now consider the impact of default risk on contracts such as swaps and forward contracts thatcan become either assets or liabilities. (As before, we assume that the transaction under considera-tion is subject to no netting, collateralization, or downgrade triggers.) We suppose that defaultscan occur only at times fy, t2, ..., t,. Define v; as value of a contract that pays off max[f*(¢;), 0] attime t; and
u; = pl — R)
where p; is the probability of default at time 1; and R is the recovery rate. From equation (26.2) anCredit Derivatives 649
appropriate expression for u; is
a, =e OED PED e-bG oli (27.6)
where y(t) and y*(t) are zero-coupon yields on corporate and risk-free bonds maturing at time t.
Given our assumption that the claim amount on a derivative equals its no-default value, u; is theloss percentage from defaults at time t;. The variable 2, is the present value of the claim made in theevent of a default at time t;. It follows that
FO) = FO) = Do urn, (27.7)i=1
Currency Swap Example
To illustrate the use of equation (27.7), suppose that a financial institution enters into a fixed-for-fixed foreign currency swap with a counterparty, whereby it receives interest in dollars and paysinterest in sterling. Principals are exchanged at the end of the life of the swap. Suppose that theswap details are as follows:
Life of swap: five yearsFPrecuency o f nayvmen te: onnua 1
Frequency of payments: annual
Sterling interest exchanged: 10% per annum (compounded annually)Dollar interest exchanged: 5% per annum (compounded annually)Sterling principal: £50 million
Dollar principal: $100 million
Initial exchange rate: 2.0000
Volatility of exchange rate: 15%
We suppose that the sterling yield curve is flat at 10% per annum (annually compounded) and thedollar yield curve is flat at 5% per annum (annually compounded) with both interest ratesconstant. We also suppose that 1-, 2-, 3-, 4-, and 5-year unsecured zero-coupon bonds issued bythe counterparty have yields that are spreads of 25, 50, 70, 85, and 95 basis points above thecorresponding riskless rate.
We assume that defaults can occur only on payment dates (i.e., just before payments are due tobe exchanged). This means that n = 5, t; = 1, t) = 2, 3 =3, ty = 4, and ts = 5. Because interestrates are assumed constant, we know that the value of the sterling bond underlying the swap ateach possible default time is 55 million pounds. Similarly, the value of the dollar bond under-lying the swap at each possible default time is 105 million dollars. The value of the swap at timet; in millions of dollars is therefore 105 — 555S(t;), where S(¢) is the dollar-sterling exchange rate attime t.
The variable »; is the value of a derivative that pays off
max{105 — 55S(t;), 0] = 55 max| 105 — S(t). o|L355 J
millions of dollars at time t;. This is 55 times the value of a foreign currency put option with astrike price of 105/55 = 1.90909. The initial exchange rate is 2.0000, the domestic interest rate is 5%650 CHAPTER 27
Table 27.2 Cost of defaults in millions of dollars on a currencyswap with a corporation when dollars are received and sterling
 
 
is paidMaturity, t; uj 0 WjU;1 0.002497 5.9785 0.01492 0.007453 10.2140 0.07613 0.010831 13.5522 0.14684 0.012647 16.2692 0.20585 0.012962 18.4967 0.2398Total 0.6834
with annual compounding (or 4.879% with continuous compounding), the foreign risk-free rate is10% with annual compounding (or 9.531% with continuous compounding), the exchange ratevolatility is 15%, and the time to maturity is t;. The v; can therefore be obtained from the softwareDerivaGem. For example, when 1; =3, the value of the option is 0.246403, so that v; =55 x 0.246403 = 13.5522.
The u,’s can be calculated from (27.6) and are shown in Table 27.2. For example,uz =e 00050%2 __ 9.00703 _ 9.010831. Table 27.2 shows the full set of w;’s and v,’s for thisexample and the calculation of the cost of default. The total cost of defaults is 0.6834 milliondollars, or 0.6834% of the principal. Table 27.3 shows a similar set of calculations for the situationwhere the financial institution is paying dollars and receiving sterling. In this case, v; is the price ofa foreign currency call option maturing at time t;. Table 27.3 shows that the expected cost ofdetaults is 0.2404 miilion dollars, or 0.2404% of the principal.
This example illustrates the general rule that a financial institution has more default risk when itis receiving a low-interest-rate currency and paying a high-interest-rate currency than the other wayround. The reason is that the high-interest-rate currency is expected to depreciate relative to thelow-interest-rate currency causing the low-interest-rate bond underlying the swap to appreciate invalue relative to the high-interest-rate bond.
The total cost of defaults on the matched pair of swaps in Tables 27.2 and 27.3 (when there aretwo different counterparties) is 0.6834 + 0.2404 = 0.9238 million dollars, or about 0.924% of theprincipal. Using a discount rate of 5% per annum, this is equivalent to payments of about
Table 27.3 Cost of defaults in millions of dollars on a currencyswap with a corporation when dollars are paid and sterling is
 
receivedMaturity, t; u; vy; UD;1 0.002497 5.9785 0.01492 0.007453 5.8850 0.04393 0.010831 . 5.4939 0.05954 0.012647 5.0169 0.06345 0.012962 4.5278 0.0587
 
Total 0.2404Credit Derivatives 651
 
Expectedexposure
            
Currencyswaps
Interest rateswaps
 
>Time
Figure 27.4 Expected exposure on a matched pair of interest rate swaps
matched terest rat ap
and a matched pair of currency swaps
0.21 million dollars per year for five years. Because the principal is 100 million dollars, 0.21 milliondollars is 21 basis points. We have therefore shown that (ignoring netting, collateralization, anddowngrade triggers) the financial institution should seek a bid-offer spread of about 21 basispoints on a matched pair of currency swaps to compensate for credit risk when the exchange ratevolatility is 15%.”
Interest Rate Swaps vs. Currency Swaps
The impact of default risk on interest rate swaps is considerably less than that on currency swaps.Using similar data to that for currency swaps, the required total spread for a matched pair ofinterest rate swaps is about 2 basis points. Figure 27.4 shows the reason for this. It compares theexpected exposure on a matched pair of offsetting interest rate swaps with the expected exposure ona matched pair of offsetting currency swaps. The expected exposure on a matched pair of interestrate swaps starts at zero, increases, and then decreases to zero. By contrast, expected exposure on amatched pair of currency swaps increases steadily with the passage of time.!°
Generalization
In reality, a financial institution is likely to have several contracts outstanding with each of itscounterparties and these are likely to be subject to a complicated set of netting, downgrade trigger,and collateralization agreements. The procedure we have outlined in this section provides aframework that can be used to estimate expected credit losses, but to be a practical tool it needsto be generalized. Suppose that a financial institution is calculating the present value of expectedlosses from defaults with a particular counterparty. The variable u; is defined as the loss percentage
° Further calculations show that the total cost of defaults, and therefore the required bid-offer spread, on amatched pair of currency swaps is relatively insensitive to the interest rates in the two currencies.
10 ‘This is largely because of the impact of the final exchange of principal in the currency swap.652 CHAPTER 27
from defaults at time t; as in Table 27.2. The variable »; must be estimated as the claim that will bemade in the event of a default at time t;. A precise estimate of v;, as in our currency swap example,is not usually possible. An approximation based on projections of the future value of the portfolioof contracts with the counterparty and the likely effectiveness of the netting, downgrade trigger,and collateralization agreements must be used.
27.6 CONVERTIBLE BONDS
 
Convertible bonds are bonds issued by a company where the holder has the option to exchange thebonds for the company’s stock at certain times in the future. The conversion ratio is the number ofshares of stock obtained for one bond. This is usually constant but is sometimes a function