When carrying out scenarioanalyses to calculate potential future losses from defaults, we should use real-world defaultprobabilities.
26.5 USING EQUITY PRICES TO ESTIMATEDEFAULT PROBABILITIES
 
 
The approaches we have examined up to now for estimating a company’s probability of defaulthave relied on the company’s credit rating. Unfortunately, credit ratings are revised relativelyinfrequently. This has led some analysts to argue that equity prices can provide more up-to-dateinformation for estimating default probabilities.
10 Note that, in this type of analysis. it is important to distinguish between the promised return on a corporate bond,which is its yield, and the expected return, which takes into account expected losses.622 CHAPTER 26
In 1974, Merton proposed a model where a company’s equity is an option on the assets of thecompany.'! Suppose, for simplicity, that a firm has one zero-coupon bond outstanding and thatthe bond matures at time T. Define:
Vy): Value of company’s assets today
V;: Value of company’s assets at time T
E,: Value of company’s equity today
Ey: Value of company’s equity at time T
D: Amount of debt interest and principal due to be repaid at time T
oy: Volatility of assets
og: Volatility of equityIf Vr < D, it is (at least in theory) rational for the company to default on the debt at time 7. Thevalue of the equity is then zero. If Vr > D, the company should make the debt repayment at time T
and the value of the equity at this time is Vr — D. Merton’s model, therefore, gives the value of thefirm’s equity at time T as
Ey = max(V; — D, 0)This shows that the equity is a call option on the value of the assets with a strike price equal to oe\,repayment required on the debt. The Black—Scholes formula gives the value of the equity today as
Ey = VoN(d;) — De™! N(d>) (26.10)whereaun Vo/D + (r +09 /2)T~~ oyvTandad, =d,— oyvT
The value of the debt today is Vy — Eo.
The risk-neutral probability that the company will default on the debt is N(—d). To calculate this,we require Vy and oy. Neither of these are directly observable. However, if the company is publiclytraded, we can observe Ey. This means that equation (26.10) provides one condition that must besalisfied by Vo and oy. We can also estimate og. From 16's lemma,
OEOr Eg =z ovo
avorapFy = N(d,)oy Vo (26.11)
This provides another equation that must be satisfied by Vo and oy. Equations (26.10) and (26.11)provide a pair of simultaneous equations that can be solved for Vy and oy.'?
Example26.2 The value of acompany’s equity is $3 million and the volatility of the equity is 80%.The debt that will have to be paid in one year is $10 million. The risk-free rate is 5% per annum. In this
" See R. Merton, “On the Pricing of Corporate Debt: The Risk Structure of Interest Rates,” Journal of Finance,29 (1974), 449-70.
'2 To solve two nonlinear equations of the form F(x, y) = 0 and G(x, y) = 0, we can use the Solyer routine in Excelto find the values of x and y that minimize [F(x, y)]’ +[G(x, y)-Credit Risk 623
 
case, Ey = 3, o¢ = 0.80, r = 0.05, T = 1, and D = 10. Solving equations (26.10) and (26.11) yieldsVo = 12.40 and oy =0.2123. The parameter d, is 1.1408, so that the probability of default isN(—d) = 0.127, or 12.7%. The market value of the debt is Vy) — Ey or 9.40. The present value ofthe promised payment on the debt is 10e~°°5”' = 9.51. The expected loss on the debt is therefore(9.51 — 9.40)/9.51 or about 1.2% of its no-default value. Comparing this with the probability of
default gives the expected recovery in the event of a default as (12.7 — 1.2)/12.7, or about 91%.
Up to now, we have assumed that all of the company’s debt is repayable at one time. In practice,debt repayments are likely to be required at a number of different times. This makes the modelrelating Vj and Ey more complicated than equation (26.10), but in principle it is still possible to usean option pricing approach to obtain estimates of V) and oy. The probability of the companydefaulting at different times in the future can then be estimated.
How well do the default probabilities produced by Merton’s model correspond to actual defaultexperience? The answer is that there is a significant difference between the two, but the defaultprobabilities produced by Merton’s model can be useful for estimating actual default probabilities.KMV, acompany in California, applies a transformation to the probabilities from Merton’s model toproduce default probability estimates.'? Moody's Risk Management Services uses the output fromMerton’s model in conjunction with other financial variables to get its default probability estimates,
This completes our discussion of the estimation of default probabilities in this chapter. In thenext chapter we will cover credit default swaps and explain how dealer quotes for credit defaultswaps provide another way to obtain default probability estimates.
26.6 THE LOSS GIVEN DEFAULT
 
 
to now in this chapter, we have focused on the estimation of the probability that a company willdefault at different future times. We now consider the estimation of the expected loss in the event thata défault occurs. This is known as the /oss given default and is often abbreviated LGD. Theprobability of default (often abbreviated PD) when multiplied by the LGD gives the expected loss.The LGD on a loan made by a financial institution is usually assumed to be
V—R(L+A)
where L is the outstanding principal on the loan, A is the accrued interest, R is the expectedrecovery rate, and V is the no-default value of the loan. The loss given default on a long position ina bond is calculated in a similar way with L equal to the face value of the bond. /
For derivatives the LGD is more complicated. This is because the claim that will be made in theevent of default is less certain than it is for a loan or a bond. We can distinguish three types ofderivatives:
1. Those that are always a liability2. Those that are always an asset3. Those that can be either an asset or a liability
Derivatives in the first category have no credit risk. If the counterparty goes bankrupt, there will be
3 To be more specific, KMV has two transformations: one to estimate real-world probabilities of default, the otherto estimate risk-neutral probabilities of default.624 CHAPTER 26
no loss. Consider, for example, a company that has written an option. This derivatives position isinitially a liability and stays a liability for the whole life of the option. If the counterparty (i.e., thebuyer of the option) goes bankrupt, the option is one of the counterparty’s assets. It is likely to beretained. closed out, or sold to a third party. In all cases, there is no loss (or gain) to the option writer.
Derivatives in the second category always have credit risk. If the counterparty goes bankrupt, aloss is likely to be experienced. Consider, for example, a company that has bought an option. Thisis initially an asset and remains an asset for the whole of the life of the option. If the counterparty(ie., the writer of the option) goes bankrupt, the option holder makes a claim and may eventuallyreceive some percentage of the value of the option.
Derivatives in the third category may or may not have credit risk. In Section 6.7 we consideredswaps. When a company first enters into an interest rate swap, it has a value equal to, or very closeto, zero. As time passes, interest rates change and the value of the swap may become positive ornegative. If the counterparty defaults when the value of the swap is positive, a claim will be madeagainst the assets of the counterparty and a loss is likely to be experienced. If the counterpartydefaults when the value of the swap is negative, no loss is made. The swap will be retained, closedout, or sold to a third party.
In Chapter 27 we will discuss the impact of credit losses on the valuation of derivatives.
Netting
A complication in the estimation of the losses that will be taken in the event of a counterpartydefault is netting. This is a clause in most contracts written by financial institutions. It states that if
a counterparty defaults on one contract with the financial institution then if must default on all
outstanding contracts with the financial institution.
Netting has been successfully tested in the courts in many jurisdictions. It can substantiallyreduce credit risk for a financial institution. Consider, for example, a financial institution that hasthree contracts outstanding with a particular counterparty. The contracts are worth +$10 million,+$36 million, and —$25 miition to ihe financial insiiiution. Suppose the counterparty runs iniofinancial difficulties and defaults on its outstanding obligations. To the counterparty, the threecontracts have values of —$10 million, —$30 million, and +$25 million, respectively. Withoutnetting, the counterparty would default on the first two contracts and retain the third for a loss tothe financial institution of $40 million. With netting, it is compelled to default on all threecontracts for a loss to the financial institution of $15 million. (If the third contract had beenworth —$45 million to the financial institution, the counterparty would choose not to default andthere would be no loss to the financial institution.)
Suppose that a financial institution has a portfolio of N derivative contracts with a particularcounterparty. Suppose that the no-default value of the ith contract is V; and the recovery made in theevent of default is R. Derivatives typically rank as unsecured claims in the event of default and theclaim made in the event of default is the no-default value at the time of default. Without netting, thefinancial institution loses
N(1 ~ R) )° max(¥;, 0)With netting, it loses =
N(1—R) max( V;, 0)i=l
Without netting, its loss is the payoff from a portfolio of call options on the contract values whereCredit Risk 625
each option has a strike price of zero. With netting, it is the payoff from a single option on theportfolio of contract values with a strike price of zero. The value of an option on a portfolio is nevergreater than, and is often considerably less than, the value of the corresponding portfolio of options.
Reducing Exposure to Credit Risk
Apart from including a netting clause, there are two other ways financial institutions can reducepotential losses in the event of a default.
The first is known as collateralization. Consider contracts entered into between two companies Aand B. Collateralization could require company A to deposit collateral with company B or with athird party. The collateralization agreement would typically state that at any given time the collateralposted should be an amount greater than the positive value (if any) that the company B has in itsoutstanding contracts with company A. Suppose that company B is the financial institutionconsidered earlier, so that it has three contracts outstanding worth +$10 million, +$30 millionand —$25 million with company A. It would require a collateral of at least $15 million. Thecollateralization agreement typically states that the collateral is adjusted periodically (e.g., everyweek) to reflect the value of the outstanding contracts. The collateral serves the same function as themargin in a futures contract and can significantly reduce credit risk. If company A is unable orunwilling to meet a demand for additional collateral, the collateralization agreement will state thatcompany B has the right to close out all outstanding contracts. Collateralization requires the twoparties to agree on a valuation model for the contract and to agree to a rate of interest paid on thecollateral. Normally a collateralization agreement is two way. This means that company A isrequired to post collateral when the value of the outstanding contracts is positive to company B andcompany B is required to post collateral when the value of the outstanding contracts is positive tocompany A.'*
The second way of reducing potential losses in the event of default is by the use of downgradetriggers, These are clauses in contracts with counterparties that state that if the credit rating of thecounterparty falls below a certain level, say A, then the contract is closed out using a pre-determined formula with one side paying a cash amount to the other side. Downgrade triggerslead to a significant reduction in credit risk, but they do not completely eliminate all credit risk. Ifthere is a big jump in the credit rating of the counterparty, say from A to default, in a short periodof time, the financial institution will still suffer a credit loss.
Sometimes financial institutions find ways to design contracts to reduce credit risk. Consider, forexample, a financial institution wishing to buy an option from a counterparty with a low creditrating. It might insist on a zero-cost package that involves the option premium being paid inarrears. This reduces the company’s exposure arising from the option position.
Diversification is an important way of managing credit risk. Most large financial institutionshave internal rules that limit the exposure they are allowed to have to a particular company, aparticular industry, or a particular country. Traders are denied the authority to enter into a newtrade if it would lead to a particular credit limit being exceeded. Credit derivatives, which will bediscussed in Chapter 27, provide a way of actively managing credit risk and are sometimes used toovercome these restrictions on the activities of traders.
'4 Long-Term Capital Management is an example of a company that managed to avoid the need for a credit ratingby entering into two-way collateralization agreements with its counterparties. After the Russian default of 1998, thevalue of its portfolio declined and it was unable to meet demands for additional collateral from its counterparties.See Chapter 30 for further discussion of the failure of Long-Term Capital Management.626 CHAPTER 26
26.7, CREDIT RATINGS MIGRATION
 
 
Over time, bonds are liable to move from one rating category to another. This is sometimesreferred to as credit ratings migration. Rating agencies produce from historical data a ratingstransition matrix. This shows the percentage probability of a bond moving from one rating to
another durine a certain nericd of time Llsually the sericd of time one Table 26.2anoiner auring a certain perioa of time. Usuauy tne perioag of time 18 one year. racic 26.6
shows data produced by S&P. It shows that a bond that starts with a BBB credit rating has an89.24% chance of still being a BBB at the end of one year. It has a 0.24% chance of defaultingduring the year, a 4.44% chance of dropping to BB, and so on. Appendix 26A shows how atable such as Table 26.8 can be used to calculate transition probabilities for time periods otherthan one year.
The probabilities in Table 26.8 are based on historical data and are therefore real-worldprobabilitics. As wo will sco later, they arc uscful for sconario analysis and the calculation ofvalue at risk. For valuing derivatives that depend on credit rating changes, we require a risk-neutraltransition matrix. This proves to be quite difficult to estimate in practice.!* One approach is basedon bond prices. To provide a simple example, we suppose that there are three rating categories, A,B, and C, and that D is used to denote default. Table 26.9 shows hypothetical data on thecumulative risk-neutral probabilities of default. This could be calculated from bond prices asdescribed in Section 26.1.
Suppose that M is a 4 x 4 matrix showing the risk-neutral transition probabilities. This meansthat M,, is the probability that an A-rated company stays A-rated; M>; is the probability that a B-rated becomes C-rated: M3q is the probability that a C-rated company defaults; and so on. Defined; as a vector that is the ith column of the matrix in Table 26.9. It follows that d,. = Md,,d, = Md, = M'd,, dy = Md, = M?d,, and d; = Md, = M‘d,. There are nine unknown ele-ments of M. These are M,,; for | <i, j < 3.!° Table 26.9 defines 15 relationships that should hold.(These are the probabilities of A-, B-, and C-rated bonds defaulting within 1, 2, 3, 4, and 5 years.)
Table 26.8 One-year transition matrix of percentage probabilities
Rating at year-end
 
Initial
rating AAA AA A BBB BB B CCC DefaultAAA 93.66 583 040 0.09 0.03 0.00 0.00 0.00AA 0.66 91.72 694 049 0.06 0.09 0.02 0.01A 0.07 2.25 91.76 518 049 0.20 0.01 0.04BBB 0.03 0.26 483 8924 444 O81 0.16 0.24BB 0.03 0.06 0.44 6.66 83.23 7.46 1.05 1.08B 0.00 «60.10 032 046 5.72 8362 3.84 5.94ccc 0.15 0.00 0.29 0.88 191 10.28 61.23 25.26
Default 0.00 0.00 0.00 0.00 0.00 0.00 0.00 100.00
Source: Standard & Poor’s, January 2001.
'S Fortunately relatively few credit derivatives have payoffs dependent on rating changes.'§ Note that Mj4=1— Mj) — Miz — M,3, and My; = | when j = 4 and zero otherwise.Credit Risk 627
Table 26.9 Cumulative percentage probabilities of default forrating categories
Lnitial I 2 3 4 5
rating year year year year yearA 0.67 1.33 1.99 2.64 3.29B 1.66 3.29 4.91 6.50 8.08Cc 3.29 6.50 9.63 12.69 15.67D 100.00 100.00 100.00 100.00 100.00
We therefore choose the nine parameters to minimize the 15 sums of the squares of the differencesbetween the elements of M‘~'d, and the corresponding elements of d; (1 <i <5) in Table 26.9.The results are shown in Table 26.10. For this example the method works well. The transitionprobabilities are plausible and the cumulative default probabilities calculated from them are closeto those in Table 26.9 (see Problem 26.13).
In practice, data on cumulative default probabilities are not always as well behaved as the datapresented in Table 26.9. This has led some researchers to suggest other methods that are based onassuming that the real-world transition matrix changes in a particular way when we move from thereal world to the risk-neutral world."’
Table 26.10 Best-fit risk-neutral transition matrix.(Probabilities expressed as percentages.)
Rating at year-end
Initial 2rating A B Cc D
A 98.4 0.9 0.0 0.7B 0.5 97.1 0.7 1.7c 0.0 0.0 96.7 3.3
Default 0.0 0.0 0.0 100.0
26.8 DEFAULT CORRELATIONS
 
 
The term default correlation is used to describe the tendency for two companies to default at aboutthe same time. There are a number of reasons why default correlations exist. Two companies in thesame industry or the same geographic region tend to be affected similarly by external events and asa result may experience financial difficulties at the same time. Also, economic conditions generallycause average default rates to be higher in some years than in others.
" See R.A. Jarrow, D. Lando, and 8. M. Turnbull, “A Markov Model for the Term Structure of Credit Spreads,”Review of Financial Studies, 10 (1997), 481-523; M. Kijfima, “A Markov Chain Model for Valuing CreditDerivatives,” Journal of Derivatives, 6, no. | (Fall 1998), 97-108.628 CHAPTER 26
Default Correlation Measures
Consider two companies A and B. A default correlation measure that is often calculated by ratingagencies is the coefficient of correlation between:
1. Avariable that equals 1 if company A defaults between times 0 and T and zero otherwise; and2. A variable that equals | if company B defaults between times 0 and T and zero otherwise
The measure is
Pas(T) — Qa(T)Oz(T)[Qa(T) — Qa(T)MOa(T) — On(TY’]
Bap(T) =
where Pa,(T7) is the joint probability of A and B defaulting between time zero and time T. Asbefore, Qa(T) is the cumulative probability that company A will default by time 7, and Q,(7) isthe cumulative probability that company B will default by time 7. Typically Bag(7T) depends on T,the length of the time period considered. Usually it increases as T increases.
Another correlation measure that is sometimes used is obtained from the probability distributionof the time to default. Suppose that tq and tg are the times to default of companies A and B. The
variable t, and fs are not normally distributed. However,
lable ty are distributed, Ho
ua(ta) = N~'[Qa(ta)l
and
un(tp) = N'[Oa(ta)]
are functions of ta and tg that are normally distributed. (As usual, N is the cumulative standardnormal distribution function.) We define the correlation measure as
Pap = corr[ua(ta), Up(tp)]
We assume that the variables ua(t,) and up(tg) have a bivariate normal distribution. This meansthat the joint probability distribution of the times to default can be described by the cumulativeprobability distribution Qa(t,) of f4, the cumulative probability distribution Op(tg) of tg, and pap.The assumption is referred to as using a Gaussian copula.
The Gaussian copula approach can be extended to the situation where there are manycompanies. Suppose there are N companies and t, is the time to default of company i. DefineQ,(t;) as the cumulative probability distribution of ¢; and
ulti) = N~'[0,(4)]
for | <i < N. We assume that the u;(t;)’s are multivariate normal.
The Gaussian copula approach is a useful way representing the correlation structure betweenvariables that are not normally distributed. It allows the correlation structure of the variables to beestimated separately from their marginal distributions. Although the variables themselves are notmultivariate normal, the approach assumes that after a transformation is applied to each variablethey are multivariate normal.
Example 26.3 Suppose that we wish to simulate defaults during the next five years in 7 companies.For each company the cumulative probability of a default during the next 1, 2, 3, 4, and 5 years is 1%,3%, 6%, 10%, and 15%, respectively. When a Gaussian copula is used, we sample from a multivariateCredit Risk 629
normal distribution to obtain u;(t) (1 < i <n). We then convert the u,(t) to t, a Sample time to default.When the sample from the normal distribution i is less than N7!(0.01) = —2.33, a default takes placewithin the first year, when the sample is between —2.33 and N~!(0.03) = ~1.88, a default takes placeduring the second year: when the sample is between —1.88 and N7'(0,06) = —1.55, a default takesplace during the third year; when the sample is between —1.55 and N~!(0.03) = —1.28, a default takesplace during the fourth year; when the sample is between —1.28 and N—'(0.03) = —1.04, a defaulttakes place during the fifth year. When the sample is greater than —1.04, there is no default.
Relationship between Measures
As in Appendix 12C, we define M(a, b; p) as the probability in a standardized bivariate normaldistribution that the first variable is less than a and the second variable is less than b when thecoefficient of correlation between the variables is o. Suppose that pag is the default correlationbetween A and B in the Gaussian copula model. Suppose also that ua(t) and up(t) are thetransformed times to default for companies A and B in the Gaussian copula model. It follows that
Pap(L) = M(ua(T), up(T); Pas)so thatM(ua(Z), un(7); Pan) — Qa(T)On(T)
lat?) — Qa(T)*UOn(7) ~ Onl7)"I
 
Ban(T) (26.12)
 
This shows that, if Q,(T) and Qp(T) are known, Bag(T) can be calculated from pap and vice versa.Usually pay is markedly greater than Bap(T).Example 26.4 Suppose that the prob bability of company A defaulting i in a one-year period is 1%and the probability of company B defaulting in a one-year period is also 1%. In this case,ua(1) = ug(1) = —2.326. If pap is 0.20, then M(ua(1), up(l), Pas) = 0.000337 and equation (26.12)shows that B,p(T) = 0.024 when T = 1.
Two alternative types of models of default correlation are known as reduced-form models andstructural models,
Reduced-form models assume that the hazard rates for different companies follow stochasticprocesses and are correlated with macroeconomic variables. When the hazard rate for company Ais high, there is a tendency for the hazard rate for company B to be high. This induces a defaultcorrelation between the two companies.
Reduced-form models are mathematically attractive and reflect the tendency for economiccycles to generate default correlations. They can be made consistent with either historical defaultprobabilities or the risk-neutral probabilities of default backed out from corporate bond prices.Their main disadvantage is that the range of default correlations that can be achieved is limited.Even when there is a perfect correlation between two hazard rates, the corresponding correlationbetween defaults in any chosen period of time is usually very low. This is liable to be a problemin some circumstances. For example, when two companies operate in the same industry and thesame country or when the financial health of one company is for some reason heavily dependenton the financial health of another company, a relatively high default correlation may bewarranted. The problem can be solved by extending the model so that the hazard rate exhibitslarge jumps.630 CHAPTER 26
Structural models are based on an idea similar to that discussed in Section 26.5. A companydefaults if the value of its assets is below a certain level. Default correlation between companies Aand B is introduced into the model by assuming that the stochastic process followed by the assetsof company A is correlated with the stochastic process followed by the assets of company B.Structural models have the advantage over reduced-form models that the correlation can be madeas high as desired. Their main drawback is that they are not consistent with the probability ofdefault calculated from historical data or bond prices.'®
26.9 CREDIT VALUE AT RISK -
 
 
The credit value-at-risk measure is defined similarly to the value-at-risk measure in Chapter 16.The credit VaR answers the question: What credit loss is such that we are X percent certain it willnot be exceeded in time 7? The time horizon T is usually longer for credit VaR than for the marketrisk VaR measure in Chapter 16. A typical time horizon for VaR when market risk is beingconsidered is ten days, whereas a typical time horizon for credit VaR is one year.
Credit losses are experienced not only when a counterparty defaults. If a counterparty’s
adit voting radi, fen A BBD «a lage jg takan Gf ofl outstanding cantracts with thecicdit rating reduces from, say, A to BBR, a loss is taken if all outstanding contracts with the
counterparty are revalued to reflect the new rating. This section sketches out two alternativeapproaches to calculating credit VaR. The first is aimed at quantifying the probabilitydistribution of the losses arising solely from counterparty defaults. The second is aimed atquantifying the probability distribution of losses arising from both credit-rating changes anddefaults. In the long run, the two types of losses should be the same, but in the short run theymay be different.!? Note that we use real-world probabilities of default when calculating credit
VaR. It is therefore appropriate to base default probabilities on historical data rather than on
bond prices.
Credit Risk Plus
Our first approach is based on a methodology proposed by Credit Suisse Financial Products in1997 and has been named Credit Risk Plus.”° It utilizes ideas that are well established in theinsurance industry.
Suppose that a financial institution has N counterparties of a certain type and the probability ofdefault by each counterparty in time T is p. The expected number of defaults for the wholeportfolio, 4, is given by « — Np. Assuming that default events are independent and p is small, theprobability of n defaults is given by the Poisson distribution as“Hy
nt
e
 
This can be combined with a probability distribution for the losses experienced on a single
*® Hull and White show how to overcome this drawback. See J. Hull and A. White, “Valuing Credit Default SwapsIk Modeling Default Correlations,” Journal of Derivatives, 8, no. 3 (Spring 2001), 12~22.
> Accountants would refer to this as a Joss recognition issue. The difference between the two approaches is really inhow losses get allocated to time periods. Of course, even though the average losses are the same under both measures,the extreme tails of the probability distribution of losses (which is what determines credit VaR) may be different.
°° See Credit Suisse Pinancial Products, “Credit Risk Management Framework,” October, 1997.Credit Risk 631
counterparty default (taking account of the impact of netting) to obtain a probability distributionfor the total default losses from the counterparties. To estimate the probability distribution forlosses from a single counterparty default, we can look at the current probability distribution of ourexposures to counterparties and adjust this according to historical recovery rates.
In practice, it is likely to be necessary for the financial institution to consider several categories
of counterparties, This means that the analysis just described must be carried out for each categoryand the results combined.
Another complication is that default rates vary significantly from year to year. Data provided byMoody’s shows that the default rate per year for all bonds during the 1970 to 1999 period rangedfrom 0.09% in 1979 to 3.52% in 1990. To account for this, we can assume a probabilitydistribution for the overall default rate based on historical data such