as that provided by Moody’s.The probability of default for each category of counterparty can then be assumed to be linearlydependent on this overall default rate.
Credit Suisse Financial Products show that, if certain assumptions are made, the total lossprobability distribution can be calculated analytically. To accommodate more general assump-tions, Monte Carlo simulation can be used. The procedure is as follows:
. Sample an overall default rate.
. Calculate a probability of default for each category.. Sample a number of defaults for each category.
. Sample a loss for each default.
A kW Ne
. Calculate total loss.6. Repeat steps | to 5 many times.
The effect of assuming a probability distribution for default rates in the way just described is tobuild in default correlations. It has the effect of making the probability distribution of total defaultlosses positively skewed, as indicated in Figure 26.2.
df ee >
0 LossFigure 26.2 Probability distribution of default losses632 CHAPTER 26
CreditMetrics
J.P. Morgan have developed a method for calculating credit VaR known as CreditMetrics. Thisinvolves estimating a probability distribution of credit losses by simulating credit rating changes foreach counterparty. Suppose we are interested in determining the probability distribution of lossesover a one-year period. On each simulation trial, we sample to determine the credit rating changesof all counterparties throughout the year. We also sample changes in the relevant market variables.We revalue our outstanding contracts to determine the total of credit losses from defaults andcredit rating changes,”!
This-approach is clearly more complicated to implement than the Credit Risk Plus approach,where only default events are modeled. The advantage is that the precise terms of outstandingcontracts can—~-at least in theory— be incorporated. Suppose, for example, that a particular contractwith an A-rated counterparty includes a downgrade trigger stating that the contract is closed outwhenever the counterparty’s credit rating drops to BBB or lower. The simulation could monitorcredit rating changes on a month-to-month basis to take this into account. It would incorporate thecondition that a loss occurs only if the credit rating changes directly from A to default.
In sampling to determine credit losses, the credit rating changes for different counterpartiesshould not be assumed to be independent. CreditMetrics suggests using a Gaussian copula (seeSection 26.7) to model the rating changes of many different counterparties. It suggests assumingthat the correlation between the normal variates corresponding to two counterparties be assumedto be the same as the correlation between their equity prices.
As an illustration of the CreditMetrics approach, suppose that we are simulating the rating changeof a AAA and a BBB company over a one-year period using the trasition matrix in Table 26.8.Suppose that the correlation between the equities of the two companies is 0.2. On each simulationtrial we would sample two variables x and y from normal distributions so that their correlation is 0.2.The variable x determines the new rating of the AAA company and the variable y determines therating of the BBB company. Because
N7'(0.9366) = 1.527
N7'(0.9366 + 0.0583) = 2.569N '(0.9366 + 0.0583 + 0.0040) = 3.062
the AAA company stays AAA-rated if x < 1.527, it becomes AA-rated if 1.527 < x < 2.569, itbecomes A-rated if 2.569 < x < 3.062, and so on. Similarly, because
N~'(0.0003) = —3.432N7'(0.0003 + 0.0026) = —2.759N7!(0.0003 + 0.0026 + 0.0483) = —1.633
the BBB company becomes AAA-rated if y < —3.432, it becomes AA-rated if —3.432 < y < —2.759,it becomes A-rated if —2.759 < y < —1.633, and so on. The appealing feature of the Gaussiancopula approach is that it allows us to use the properties of multivariate normal distribution tojointly sample the credit migration of many companies.
 
2 We determine losses from credit rating changes by valuing the contract at the end of the year with (i) the creditrating at the beginning of the year and (ii) the credit rating at the end of the year. The credit loss is the differencebetween the two.Credit Risk 633
SUMMARY
 
 
There are a number of different ways of estimating the probability that a company will defaultduring a particular period of time in the future. One involves bond prices, another involveshistorical data, and a third involves equity prices. The default probabilities backed out from bondprices are risk-neutral probabilities. The probabilities backed out from historical data are real-world probabilities. Real-world probabilities should be used for scenario analysis and thecalculation of credit VaR. Risk-neutral probabilities should be used for valuing credit-sensitiveinstruments. In general risk-neutral default probabilities are significantly higher than real-worldprobabilities.
The expected loss experienced from a counterparty default is reduced by what is known asnetting. This is a clause in most contracts written by financial institutions that states that if acounterparty defaults on one contract it has with a financial institution then it must default on allcontracts it has with that financial institution. Losses are also reduced by collateralization anddowngrade triggers. Collateralization requires counterparties to post collateral, and downgradetriggers lead to contracts being closed out when the counterparty’s credit rating falls below aspecified level.
A credit VaR measure can be defined as the credit loss that, with a certain probability, will notbe exceeded during a certain time period. The credit VaR can be defined to take into account onlylosses arising from defaults. Alternatively, it can be defined so that it reflects the impact of bothdefaults and credit rating changes.
SUGGESTIONS FOR FURTHER READING
 
 
Altman, E. I., “Measuring Corporate Bond Mortality and Performance,” Journal of Finance, 44 (1989),902-22.
Black, F., and J. Cox, “Valuing Corporate Securities: Some effects of Bond Indenture Provisions,”Journal of Finance, 31, no. 2 (1976), 351-67.
Credit Suisse Financial Products, “Credit Risk Management Framework,” October 1997.
Diffie TD and K Cinalatan “Modeling Term Ctensume, ., and ww. singreton, “Moaeing .erm str
Studies, 12 (1999), 687-720.
Finger, C. C., “A Comparison of Stochastic Default Rate Models,” Risk Metrics Journal, 1 (November2000), 49-73.
Group of Thirty, “Derivatives: Practices and Principles,” Washington, DC, 1993.
Hull, J. C., and A. White, “Valuing Credit Default Swaps I: No Counterparty Default Risk,” Journal ofDerivatives, 8, no. 1 (Fall 2000), 29-40.
Kijima, M., “A Markov Chain Model for Valuing Credit Derivatives,” Journal of Derivatives, 6,no. | (Fall 1998), 97-108.
Jarrow, R. A., D. Lando, and S. M. Turnbull, “A Markov Model for the Term Structure of CreditSpreads,” Review of Financial Studies, 10 (1997), 481-523.
J.P. Morgan, “CreditMetrics Technical Document,” April 1997.
Litterman, R., and T, Iben, “Corporate Bond Valuation and the Term Structure of Credit Spreads,”Journal of Portfolio Management, Spring 199], 52-64.
 
la 3”? of Fi iade nas, Review of Mranciat634
CHAPTER 26
Merton, R. C., “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal ofFinance, 2 (1974), 449~70.
Rodriguez, R. J., “Default Risk, Yield Spreads, and Time to Maturity,” Journal of Financial andQuantitative Analysis, 23 (1988), 111-17.
QUESTIONS AND PROBLEMS(ANSWERS IN SOLUTIONS MANUAL)
 
26.4.26.5.
26.6.
26.7.
26.9.
26.10.
26.11.
26.12.
 
. Suppose that the spread between the yield on a three-year zero-coupon corporate bond and the
yield on a similar risk-free bond is 50 basis points. The corresponding spread for six-year bondsis 80 basis points. Assume a zero recovery rate. What is the risk-neutral probability of defaultbetween three and six years in a risk-neutral world?
. In Figure 26.1 the gradient of the curve for a Baa company is much steeper than that for a
Aaa company. Explain the point made in footnote 3 that this is consistent with the comparativeadvantage argument.
. Explain the difference between a risk-neutral and real-world default probability. Which is
higher? Which should be used for (a) valuation and (b) scenario analysis?Explain the difference between a hazard rate and a default probability density.
Suppose that the risk-neutral default probability for a one-year zero-coupon bond is 1% and thereal-world default probability for the bond is 0.25%. The risk-free yield curve is flat at 6% andthe excess return of the market over the risk-free rate is 5%. What does this imply about thebond’s beta? Assume no recovery in the event of default.
Show that the default probabilities for year 1 and year 2 in Table 26.5 are correct for (a) the casewhere the claim equals the no-default value and (bh) the case where the claim equals the face valueplus accrued interest.
Describe how netting works. A bank already has one transaction with a counterparty on itsbooks. Explain why a new transaction by a bank with the counterparty can have the effect ofincreasing or reducing the bank’s credit exposure to the counterparty.
. Suppose that the measure 6,g(7) in Section 26.7 is the same in the real world and the risk-
neutral world. Is the same true of the Gaussian copula measure, pap?
Explain the difference between Credit Risk Plus and CreditMetrics as far as the following areconcerned: (a) the definition of a credit loss and (b) the way in which default correlation ismodeled.
Suppose that the probability of company A defaulting during a two-year period is 0.2 and theprobability of company B defaulting during this period is 0.15. If the Gaussian copula measureof default correlation is 0.3, what is Bag(2)?
Suppose that the LIBOR curve is flat at 6% with continuous compounding and a three-yearbond with a coupon of 5% (paid semiannually) sells for 90.00. How would an asset swap on thebond be structured? What is the asset swap spread that would be calculated in this situation?
Show that the value of a coupon-bearing corporate bond is the sum of the values of itsconstituent zero-coupon bonds when the claim amount is the no-default value of the bond, butthat this is not so when the claim amount is the face value of the bond plus accrued interest.Credit Risk 635
26,13.
26.14.
26.15.
26.16.
Calculate the cumulative percentage probabilities of default for the transition matrix inTable 26.10.
ASSIGNMENT QUESTIONS
 
The value of a company’s equity is $4 million and the volatility of its equity is 60%. The debtthat will have to be repaid in two years is $15 million. The risk-free interest rate is 6% perannum. Use Merton’s model to estimate the expected loss from default, the probability ofdefault, and the recovery rate in the event of default. Explain why Merton’s model gives a highrecovery rate. (Hint: The Solver function in Excel can be used for this question.)
Use equation (26.3) to show that
jt! jn} jalGj- Yi pe; = ay(1 ~ En) SBS G;- Spies;i=l i=] isl
What limits does this inequality impose on the price and yield of a 20-year bond in the examplein Table 26.4 when the claim amount is the face value plus accrued interest?
Suppose that the risk-free rate is flat at 6% with annual compounding. One-, two-, and three-year bonds yield 7.2%, 7.4%, and 7.6% with annual compounding. All pay 6% coupons.Assume that defaults can happen only at the end of a year (immediately before a couponpayment date). Estimate the risk-ncutral probability of default at the end of each year. Assume arecovery rate of 40% and that the claim amount equals the face value plus accrued interest.636 CHAPTER 26
APPENDIX 26A
Manipulation of the Matrices of Credit Rating Changes
Suppose that A isan N x N matrix of credit rating changes in one year. This is a matrix such as theone shown in Table 26.8. The matrix of credit rating changes in m years is A”. This can be readilycalculated using the normal rules for matrix multiplication.
The matrix corresponding to a shorter period than one year, say six months or one month ismore difficult to compute. We first use standard routines to calculate eigenvectors x), x2,..., Xyand the corresponding eigenvalues Aj, A2,..., Ay. These have the property that
Ax; = Ax;
Define X as a matrix whose ith row is x;, and A as a diagonal matrix where the ith diagonalelement is A;. A standard result in matrix algebra shows that
A=X'AXFrom this it is easy to see that the nth root of A isXAXwhere A* is a diagonal matrix where the ith diagonal element is 4}/".Some authors such as Jarrow, Lando, and Turnbull prefer to handle this problem in terms of
what is termed a generator matrix.” This is a matrix T such that the transition matrix for a shortperiod of time dt is 1+ dt and the transition matrix for longer period of time, f, is
eo kexp(tl) = ak=0 .
2 See R. A. Jarrow, D. Lando, and S. M. Turnbull, “A Markov Model for the Term Structure of Credit Spreads,”Review of Financial Studies, 10 (1997), 481-523.CHAPTER 27
CREDIT DERIVATIVES
Credit derivatives are contracts where the payoff depends on the creditworthiness of one or morecommercial or sovereign entities. In this chapter we explain how the most popular credit derivativeswork and discuss how they can be valued. We build on the ideas introduced in Chapter 26concerned with default probabilities, recovery rates, and default correlations.
The chapter starts by providing a detailed discussion credit default swaps. These are the mostpopular credit derivatives. They provide a market in which default insurance can be bought andsold. The chapter then moves on to discuss a number of other types of credit derivatives: totalreturn swaps, credit spread options, and collateralized debt obligations. After that, it provides ageneral framework for taking credit risk into account in the pricing of derivatives. The final part ofthe chapter covers the pricing of convertible debentures. Convertible debentures are not usuallythought of as credit derivatives, but it turns out to be important to take the possibility of the issuerdefaulting into account in some way when a convertible debenture is valued.
27.1. CREDIT DEFAULT SWAPS
 
 
A credit default swap (CDS) is a contract that provides insurance against the risk of a default byparticular company. The company is known as the reference entity and a default by the company isknown as a credit event. The buyer of the insurance obtains the right to sell a particular bondissued by the company for its par value when a credit event occurs. The bond is known as thereference obligation and the total par value of the bond that can be sold is known as the swap’snotional principal.
The buyer of the CDS makes periodic payments to the seller until the end of the life of the CDSor 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 requirephysical delivery, the swap buyer delivers the bonds to the seller in exchange for their par value.When there is cash settlement, the calculation agent polls dealers to determine the mid-marketprice, Z, of the reference obligation some specified number of days after the credit event. The cashsettlement is then (100 — Z)% of the notional principal.
An example may help to illustrate how a typical deal is structured. Suppose that two partiesenter into a five-year credit default swap on March 1, 2002. Assume that the notional principal is$100 million and the buyer agrees to pay 90 basis points annually for protection against default bythe reference entity. If the reference entity does not default (i.e., there is no credit event), the buyerreceives no payoff and pays $900,000 on March | of each of the years 2003, 2004, 2005, 2006,
637638 CHAPTER 27
Default 90 basis points per year Default>protection protection
buyer Payment if default by sellerreference entity
 
 
 
 
Figure 27.1 Credit default swap
and 2007. If there is a credit event, a substantial payoff is likely. Suppose that the buyer notifies theseller of a credit event on September |, 2005 (halfway through the fourth year). If the contractspecifies physical settlement, the buyer has the right to sell $100 million par value of the referenceobligation for $100 million. If the contract requires cash settlement, the calculation agent wouldpoll dealers to determine the mid-market value of the reference obligation a predesignated numberof days after the credit event. If the value of the reference obligation proved to be $35 per $100 ofpar 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 accruedbetween March 1, 2005, and September 1, 2005 (approximately $450,000), but no further paymentswould be required. ‘his example is illustrated in Figure 27.1.
Table 27.1 shows quotes for credit derivatives as they might have been provided by a marketmaker in January 2001. The second column of the table shows the Moody’s and S&P credit ratingsfor the company (see Section 26.1 for a discussion of credit ratings). The last four columns showbid and offer quotes in basis points for credit default swaps with maturities of 3, 5, 7, and 10 years.For Toyota, the market maker is prepared to buy three-year default protection for 16 basis pointsper year and sell three-year default protection for 24 basis per year; it is prepared to buy five-yearprotection for 20 basis points per year and sell five-year protection for 30 basis points per year; andSo on.
Credit derivatives allow companies to manage their credit risk actively. Suppose that a bank hadseveral hundred million dollars of loans outstanding to Enron in January 2001 and was concernedabout its exposure. It could buy a $100 million five-year CDS on Enron from the market maker inTable 27.1 for 135 basis points or $1.35 million per year. This would shift to the market makerpart of the bank’s Enron credit exposure.
Instead of getting md of its credit exposure, the bank might want to exchange part of the exposurefor an exposure to a company in a totally different industry, say Nissan. The bank could sell a five-year $100 million CDS on Nissan for $1.25 million per year at the same time as buying a CDS onEnron. The net cost of this strategy would be 10 basis points or $100,000 per year. The strategy
Table 27.1 Credit default swap quotes (basis points)
 
 
MaturityCompany Rating 3 years Syears 7 years 10 yearsToyota Motor Corp Aal/AAA 16/24 20/30 26/37 32/53Merrill Lynch Aa3/AA— 21/41 40/55 41/83 56/96Ford Motor Company A+/A 59/80 85/100 95/136 = 118/159Enron Baal/BBB+ 105/125 115/135 117/158 =: 182/233
Nissan Motor Co. Ltd. Bal/BB+ 115/145 125/155 = 200/230 = 244/274Credit Derivatives 639
 
shows that credit default swaps can be used to diversify credit risk as well as to shift credit risk toanother company.
As it happens, Enron defaulted within 12 moriths of January 2001. Either the risk-shifting or therisk-diversification strategy would have worked out very well!
VaiuationWe now move on to consider the valuation of a credit default swap. For convenience we assumethat the notional principal is $1. We assume that default events, interest rates, and recovery rates
are mutually independent. We also assume that the claim in the event of default is the face valueplus accrued interest. Suppose first that default can occur only at times ¢), ty,..., t,- Define:
T: Life of credit default swap in years
p;: Risk-neutral probability of default at time ¢;
R: Expected recovery rate on the reference obligation in a risk-neutral world (this is assumedto be independent of the time of the default)
u(t): Present value of payments at the rate of $1 per year on payment dates between time zeroand time ¢
e(t) : Present value of a payment at time t equal to ¢ — * dollars, where i* is the payment dateimmediately preceding time t (both ¢ and t* are measured in years)
v(t) : Present value of $1 received at time tw: Payments per year made by credit default swap buyer per dollars: Value of w that causes the credit default swap to have a value of zerox: The risk-neutral probability of no credit event during the life of the swap
A(t): Accrued interest on the reference obligation at time ¢ as a percent face value
The value of z is one minus the probability that a credit event will occur. It can be calculated from
Le oeAe Pi-nn=1 ->p,j=1
o
The payments last until a credit event or until time 7, whichever is sooner. The present value of thepayments is therefore
w > Tutt;) + etlp: + wru(t)
i=l
If a credit event occurs at time ¢;, the risk-neutral expected value of the reference obligation, as apercent of its face value, is [1 + A(t,)]}R. The risk-neutral expected payoff from the CDS istherefore!
1-[]+AQ)]R=1-R- ACHR
' As shown by Table 26.3, rating agencies quote recovery rates as a percent of face value rather than face valueplus accrued interest. Using this (less precise) definition, the risk-neutral expected payoff is 1— R rather than
1—R-A(t)R.640 CHAPTER 27
The present value of the expected payoff from the CDS is
” a n
Soll = R= AGRI pio(t,)
i=land the value of the credit default swap to the buyer is the present value of the expected payoffminus the present value of the payments made by the buyer:
SU = R= AG) Alpwtt,) — w Sle) + elt pi + wu)
ni=l is
The CDS spread, s, is the value of w that makes this expression zero:
— EL = R= Aw) Ripe)~ lee) ett a; + u(T) (27-1)
The variable s is referred to as the credit default swap spread, or CDS spread. It is the payment peryear, as a percent of the notional principal, for a newly issued credit default swap. (For example, inTable 27.1 the mid-market value of s for Nissan for a seven-year CDS is 215 basis points, or$0.0215 per dollar of principal.)
OQZI> collar of principal
Example27.1 Suppose that the risk-free rate is 5% per annum with semiannual compounding andthat, in a five-year credit default swap where payments are made semiannually, defaults can take placeat the end of years 1, 2, 3, 4, and 5. The reference obligation is a five-year bond that pays a couponsemiannually of 10% per year. Default times are immediately before coupon payment dates on thisbond. Assume that the probabilities of default are as in Table 26.5 (final column) and the expectedrecovery rate is 0.3. In this case, A(t;) = 0.05 and e(¢;) = 0 for alli. Also, v(t,) = 0.9518, v(t.) = 0.9060,u(t3) = 0.8623, v(t4) = 0.8207, and volts) = 0.7812, while u(t) = 0.9637, u(ta) = 1.8810, u(t,) = 2.7541,u(tq) = 3.5851, and u(ts) = 4.3760. From Table 26.5, it follows that p,; = 0.0224, p, = 0.0247,Ps = 0.0269, ps = 0.0291, ps = 0.0312, and w = 0.8657. The numerator in equation (27.1) is
(1 — 9.3 — 0.05 x 0.03)x (0.0224 x 0.9518 + 0.0247 x 0.9060 + 0.0269 x 0.8623 + 0.0291 x 0.8207 + 0.0312 x 0.7812)or 0.07888. The denominator is0.0224 x 0.9637 + 0.0247 x 1.8810 + 0.0269 x 2.7541+ 0.0291 x 3.5851 + 0.0312 x 4.3760 + 0.8657 x 4.3760or 4.1712. The CDS spread, s, is therefore 0.07888/4.1712 = 0.1891, or 189.1 basis points. Thismeans that payments equal to 0.5 x 1.891 = 0.09455% are made every six months.
We can extend the analysis to allow defaults at any time. Suppose that g(t) is the risk-neutraldefault probability density at time r. Equation (27.1) becomes
IL R- AMR qe) at
fy alu + e()] at + mu(T)
Example 27.2 Consider again the situation in Example 27.1. Suppose that default can take placeat any time and the default probability density is as in Table 26.6 (final column). Equation (27.2)
gives the swap spread as 0.01944, or 194.4 basis points. (This means that payments equal to 0.972%of the principal are required every six months.)
(27.2)Credit Derivatives 641
Approximate No-Arbitrage Arguments
There is an approximate no-arbitrage argument that can be used to understand the determinantsof s. If an investor buys a T-year par yield bond issued by the reference entity and buys a T-yearcredit default swap, the investor has eliminated most of the risks associated with default by thereference entity. Suppose that the par yield is y. If there is no default by the reference entity, theinvestor realizes a return of y—ys for T years and receives a cash flow equal to the bond’s parvalue at the end of the T years. If there is a default, a return of y —s is realized until the time ofthe default. There is then a payoff on the CDS so that the value of the investor’s position isbrought to par.
The argument just given suggests that, by combining the T-year par yield bond issued by thereference entity with a 7-year CDS, the investor obtains a risk-free return of y — s. Suppose that xis the par yield on a T-year risk-free bond. For no arbitrage, we should have y — 5 = x ors = y—x.However, a close analysis of the argument shows that the arbitrage is less than perfect for thefollowing reasons:
1. The credit default swap provides a payoff of par minus the post-default value of the bond.For the arbitrage to work perfectly, it would have to provide par plus accrued interest minusthe post-default value of the bond.
2. For the arbitrage to work perfectly, the investor would have to be able to earn a risk-free rateof exactly x between a default and time 7. If the initial term structure is not flat or if interestrates are stochastic, the interest rate is liable to be different from x.
Define s* = y— x. Aiso define a as the average accrued interest on the reference bond (Le., a is theaverage value of A(t)) and a* as the average accrued interest on a par yield bond issued by thereference entity. Hull and White show that”*. - -s‘(l1—R-—aRa Stl Raa) (73)(= RY 40%)
provides a better estimate of s than s*.
Example 27.3 Consider again the CDS in Example 27.2. In this case, the reference obligation paysa 10% coupon semiannually, so that A(t) varies from 0 to 0.05 and a = 0.025. The five-year par yieldfor bonds issued by the reference entity is 7%, so that a* =0.0175. Also, R = 0,3, x =0.05, andy = 0.07, so that s* = 0.02. Equation (27.3) gives
_ 9.02(1 — 0.3 — 0.025 x 0.3)
**~4 —0.3\1 + 0.0175) 0.01945
or 194.5 basis points. This is very close to the 194.4 basis points calculated in Example 27.2.
Implying Default Probabilities from CDS Swaps
The credit default swap market is now so liquid that many analysts use it to calculate implied defaultprobabilities. This is analogous to using options markets to calculate implied volatilities fromoptions prices.
Suppose that the CDS spreads for maturities t), to,..., t, ATe S,, 52,..., 5,. We assume a stepfunction for the default probabilities and define g; as the default probability density between
?