Credit Risk. João Pedro Pereira. Teaching notes. Nova School of Business and Economics Universidade Nova de Lisboa

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1 Credit Risk Teaching notes João Pedro Pereira Nova School of Business and Economics Universidade Nova de Lisboa March 27, 2018

2 Contents 1 Foundations for Credit Risk Modelling Default Loss Exposure Sources Mitigation Loss Given Default Probability of Default Portfolio Default Loss The Default Loss Distribution Expected Loss Unexpected Loss Estimation of Default Probabilities Agency Credit Ratings Rating classes Transition Matrices and Credit Migration Drawbacks of ratings Credit Scoring and Internal Rating Models Z-Score Z -Score Z -Score Estimating the probability of default Application: loan pricing Structural approach to credit risk Merton s model Assumptions Probability of Default Price of a corporate bond Default Spread Moody s-kmv EDF Overview Estimation of the assets process

3 Contents Distance to Default Probability of Default (EDF) Summary of Differences to Merton EDF versus Agency Ratings Alternative estimation procedures Portfolio Models Credit Migration Approach CreditMetrics Assumptions Single exposure risk Portfolio risk Other models CreditPortfolioView Moody s-kmv Portfolio Manager Reduced-Form Approach CreditRisk Conclusion Valuing defaultable bonds Credit spreads Alternative spreads Pricing Pricing with objective probabilities of default Risk-neutral probabilities of default Definition Estimation from corporate bonds Risk-neutral versus objective probabilities Default intensity Risk-neutral pricing Asset Swaps Definition Asset Swap pricing Credit Derivatives Introduction Purpose Credit Event Market characteristics Credit Default Swaps Definition Usage Indexes CDS pricing Triangular Arbitrage Pricing CDS with Asset Swaps

4 Contents CDS spread vs. Z-spread Pricing with a reduced-form model Market-implied default rates Mark-to-Market of an existing CDS Standardization and upfront payments Credit Spread Options Total Return Swaps Credit-Linked Notes Collateralized Debt Obligations Introduction Assets and Liabilities Cash Flow Waterfall Synthetic CDOs Applications of CDOs Bibliography 124

5 Chapter 1 Foundations for Credit Risk Modelling 1.1 Default Loss Definition 1.1.1: Credit Risk Credit Risk is the possibility that a counterparty defaults on a payment (usually related to loans or bonds). When the counterparty defaults, we suffer a credit or default loss. To analyze this loss in detail, we need the following notation. Consider a set of i = 1,...,I obligors. Define: N i (t): Default indicator process. N i (t) = 1 if obligor i has defaulted by time t, and 0 otherwise. E i (t): Exposure process. E i (t) is the amount we would loose if obligor i defaulted at time t with zero recovery. Also called exposure at default (EAD). L i (t): Loss given default (LGD) of obligor i at time t. Have 0 LGD 1, but LGD is often less than one since a fraction of E i (t) may be recovered in bankruptcy proceedings. The Recovery Rate is given by R i (t) = 1 L i (t). Given these 3 processes, we can now define the default loss as follows. 5

6 1.2. Exposure 6 Definition 1.1.2: Default Loss The default loss for obligor i is D i (T) := N i (T) E i (τ i ) L i (τ i ) (1.1) where T is some time horizon, say 1 year, and τ i is the time of default. 1.2 Exposure One of the components of the default loss D i defined in (1.1) is the credit exposure amount E i (also called Exposure at Default, EAD). The credit exposure is the amount we would loose if the obligor defaulted with zero recovery Sources Some sources of exposure are the following: Fixed exposures. These arise from lending directly to an obligor or buying bonds. 1 Committed and unused lines of credit. These are hard to measure. Usually, the borrower will be able to draw at least part of the line of credit before his financial problems become known to the bank. A common practice is to consider a fixed fraction of a committed line of credit as EAD (even if it is not drawn at the moment). Variable exposures. These arise from over-the-counter (OTC) transactions in derivatives. Current, pre-settlement exposure is defined as the current replacement value of the contract (if this value is positive to us): E(t) = max{0, Replacement value at t} 1 The legal claim does not necessarily match the market value. Usually, for loans and bonds only the notional amount plus accrued interest are considered a legal claim; the actual market value of the bond and future coupons are not legal claims.

7 1.2. Exposure 7 Future exposure, ie, the Exposure at Default (EAD), is harder to quantify as the future value of a derivative will depend on future unknown market variables. Since Basel II, banks can compute the EAD under three different methods of increasing sophistication: Current Exposure Method (CEM), Standardized Method, and Internal Model Method (IMM). The simplest CEM defines the exposure at default as E(τ) = Replacement value at t+ Add-on The Add-on is calculated as the product of the transaction notional and the Add-on Factor, which is determined from the regulatory tables on the basis of the remaining maturity and the type of underlying instrument (e.g., interest rates, foreign exchange, etc.). The most sophisticated method, IMM, allows banks to use their own internal risk-management models to compute the EAD (the models must be approved by the supervisory authority). Typically, one uses Monte Carlo simulations to generate many possible paths of exposure values (for example, for an IRS, we could use a Hull-White model of the short rate to generate possible paths for the term structure and compute paths for the value of the IRS). Then, average the exposures at each date and take the maximum across the time horizon. The EAD is equal to this maximum times a multiplier (1.4) that accounts for model uncertainty. See Pykhtin and Zhu (2006) for details. Note that Futures are exchange traded and thus virtually credit-risk free Mitigation Several techniques can be used to mitigate the exposure: Netting Agreements. For example, when a derivative involves payments by both contract parties, a netting agrement causes the exposure to be only to the net value of the payments. In the aggregate, the presence of netting agreements reduces the gross credit exposure of all OTC derivatives contracts to 28% of their total market value (BIS data, Dec 2003).

8 1.3. Loss Given Default 8 Collateral. The counterparty more likely to default (eg, hedge fund) deposits assets (typically, liquid securities) with the other counterparty (eg, investment bank). According to International Swaps and Derivatives Association s data, about half of the aggregate credit exposure in 2003 (after netting) was covered by collateral. Limits. The credit committee of a bank may fix a maximum amount of exposure to a given counterparty. Termination rights. One party may reserve the right to terminate the contract if the credit rating of the other party worsens significantly. This may allow a better recovery of the exposure amount before actual default happens. Third-party guarantees. Similar effect to collateral. Credit derivatives. 1.3 Loss Given Default The loss given default L i, or the recovery rate R i = 1 L i, is an important factor in the default loss D i defined in (1.1). Conceptually, the recovery rate is simple. If we receive a 40 cent cash settlement for each dollar of Exposure (E i ), then the recovery rate is R i = 40% and L i = 60%. However, recovery rates are very hard to estimate in practice. This is due to several factors. First, data on recovery rates is fragmented and unreliable. Second, recovery rates depend on the bankruptcy procedure and thus on the legal environment of each country. In general, we expect recovery rates on debt to be influenced by the following factors: Collateral and Priority class (senior, junior). Secured bonds are backed by collateral (property, equipment, or other securities). Secured creditors have the first claim on their collateral, unsecured creditors come next, and stockholders are residual claimants. Junior or subordinated debt is not paid until prior senior debt is paid in full. Usually, creditors of the same legal claim amount and the same priority class are treated identically. Legislature where bankruptcy takes place. E.g., higher recovery rates in UK, lower in US and France.

9 1.3. Loss Given Default 9 Industry group of the obligor. Capital intensive business have higher recovery rates; dotcom s recovery rates are close to zero. Obligor s rating prior to default. If the obligor spent a long time close to default, it has fewer assets to liquidate and pay off creditors. The business cycle. Recovery rates tend to be lower in recessions. Hence, recessions are doubly bad: more defaults than usual; lower recovery rates than average. 2 Renaul and Scaillet (2004) estimate the recovery rates in Table 1.1 for US Corporate Bonds. Note the huge standard deviations. Unfortunately, since recovery rates depend systematically on the business cycle, there is little hope of diversifying away the estimation errors. Hence, it is hard to say what recovery rates we should input in credit risk models. 3 Table 1.1: Recovery rates for US Corporates Seniority Mean (%) Standard Deviation (%) Senior secured Senior unsecured Subordinated Junior subordinated Total Given all this uncertainty, one popular model for random recovery rates is the beta distribution. For random variables within [0, 1], the beta density is: f(x) = cx a 1 (1 x) b 1 where c = Γ(a+b) is a normalization constant. This density can assume Γ(a)Γ(b) a variety of shapes ) by changing the two parameters ) a and b. We can use a = µ 1 and b = (1 µ) 1 to fit the empirical mean ( µ(1 µ) σ 2 ( µ(1 µ) σ 2 µ and standard deviation σ. Figure 1.1 illustrates the beta densities for two bond classes from table SeeMoodys.comforcurrentestimatesofthenegativecorrelationbetween defaultrates and recovery rates. See also the survey in Altman, Resti, and Sironi (2003). 3 See Smithson (2003) for more data on recovery rates. 4 Thanks to Pedro Baltazar for correcting an error in a previous version of these notes.

10 1.4. Probability of Default 10 Figure 1.1: Beta Densities Beta densities consistent with empirical µ and σ Senior Secured Junior Subordinated Recovery Rate 1.4 Probability of Default This section discusses the third component of the default loss: the probability of default (PD). The probability of default determines the N i (T) process in (1.1). Notation: PD i (t,t) := Probability of Default of obligor i from time t until time T For instance, PD i (0,1 year) = 2% means that there is a 0.02 chance of default during the next year. Consider a single obligor. To simplify the notation, define p t := PD i (t 1, t). The possibility of default can be represented by a binomial tree. One period from now, the firm can be in one of two states: state of default (D 1 ) with probability p 1 ; or state of survival (S 1 ) with probability 1 p 1. This setup is easily extended to multiple periods, as in figure 1.2.

11 1.4. Probability of Default 11 Figure 1.2: Multi-period default tree Terminology: Marginal default probabilities denote single-period probabilities. For example, PD(2,3) is the probability of defaulting in year 3. This is also called the conditional default probability, since it is the probability of defaulting in period 3, given that the firm survived for the first 2 years: P[default in (2,3) survive in (0,2)] = P[default in (2,3) survive in (0,2)]/P[survive in (0,2)] = (1 p 1)(1 p 2 )p 3 (1 p 1 )(1 p 2 ) = p 3 = PD(2,3). Cumulative default probabilities denote multi-period probabilities. For example, PD(0, 3), is the probability of defaulting at any point until time 3. Notation: PS i (t,t) := Probability of Survival of obligor i from time t until time T is: For example, the cumulative survival probability from now until period 3 PS(0,3) = (1 p 1 )(1 p 2 )(1 p 3 )

12 1.5. Portfolio Default Loss 12 Note that there are many paths that lead to default, whereas only through one path does the firm remain alive. Hence, PD(0,3) = 1 PS(0,3) Example Given a constant one-year probability of default of 20%, the cumulative probability of default over two years is P[default 2 years] =... = 0.36 The Term Structure of default probabilities is the set of(either cumulative or marginal) default probabilities for future time periods. 1.5 Portfolio Default Loss Since a bank holds a portfolio of credits (instead of a single individual credit), the relevant random variable to be managed is the Portfolio Default Loss. Proposition 1.5.1: Portfolio Default Loss The default loss on a portfolio with I obligors is D(T) = I D i (T) = i=1 I N i (T) E i (τ i ) L i (τ i ) (1.2) i= The Default Loss Distribution The full portfolio credit loss distribution can be represented by F(x) := P[D(T) x]. This distribution will depend crucially on the dependency between individual losses. Assuming independence between individual defaults almost always leads to a gross underestimation of the portfolio s credit risk. Figure 1.3 shows the density function of the portfolio default loss (D) for a typical loan portfolio. Its main characteristics are:

13 1.5. Portfolio Default Loss 13 Not symmetrical. While the loss is truncated at zero, D 0 (best case is no loss), it can be extremely large. Highly positively skewed. Most mass in the low loss area, but long tail to the right. Heavy right tail. Very large losses are still likely. Hence, the usual VaR quantiles are associated with quite large losses. Figure 1.3: Density function of the portfolio default loss Density Portfolio Default Loss Expected Loss Let PD i (0,T) be the probability of default of obligor i until time T. We have that E[N i (T)] = PD i (0,T). Proposition 1.5.2: Expected Individual Loss Assuming that E i and L i are known and constant, the expected loss on an individual obligor over the horizon T is E[D i (T)] = PD i (0,T) E i L i (1.3) For example, if PD i (0,T) = 0.02%, E i = 100M$ and L i = 0.9, then E[D i (1)] = = 0.018M$.

14 1.5. Portfolio Default Loss 14 However, the realized loss can only be one of two possible outcomes: D i (1) = 0 (no default; very likely) or D i (1) = = 90M$ (default; unlikely). Hence, we should be careful in interpreting the expected loss for a single obligor. While the individual expected losses are small, these losses accumulate when we consider a portfolio of many obligors. From (1.2), we have the expected loss on a portfolio: Proposition 1.5.3: Expected Portfolio Loss The Expected Portfolio Loss is E[D(T)] = I E[D i (T)] i=1 Example Consider the following portfolio: Exposure (e) Prob of Default Loan A Loan B Assuming a recovery rate of 40%, the expected loss on the portfolio is E[D(T)] =... = Suffering the expected loss on a portfolio is not bad luck: it is what you should expect to happen. Thus, the portfolio s earnings (the loan s expected gains) should be enough to cover the expected loss. The principle is similar to any insurance. For example, in health insurance the cost of a few sick customers are paid by the fees collected from many healthy customers. Similarly, the sum of the (credit) risk premiums charged in each loan should be enough to cover the expected portfolio (credit) loss Unexpected Loss As a measure of an unanticipated worst-case scenario, we now define the unexpected loss. It is usually defined with respect to a VaR quantile. Let D k

15 1.5. Portfolio Default Loss 15 represent the portfolio s k VaR quantile, that is, the maximum default loss with k confidence. For example, D 99% is given by P[D(T) D 99% ] = Definition 1.5.1: Unexpected Portfolio Loss The unexpected loss of the portfolio (UEL) is the difference between the VaR quantile and the expected loss of the portfolio: UEL := D k E[D(T)] Note that this is not the maximum possible loss, which is suffered when the total portfolio defaults with zero recovery. As opposed to expected loss, the portfolio unexpected loss is not additive in the individual unexpected losses. Hence, the UEL is defined directly on the portfolio loss distribution. Also contrary to the expected loss, the unexpected loss is frequently used to determine the capital reserves that have to be held against the credit risk of the portfolio. Hence, 1. Losses up to the expected loss are borne by individual business lines; 2. Further losses above the expected loss and below the unexpected loss are borne by the capital reserves; 3. Further losses above the unexpected loss may lead to default of the bank itself. (This event can be controlled by setting the original VaR level).

16 Chapter 2 Estimation of Default Probabilities This chapter discusses different methods to estimate default probabilities. The KMV model, which is also a very important method of estimating default probabilities, will be discussed later. 2.1 Agency Credit Ratings Rating classes Public rating agencies such as Standard and Poor s, Moody s, and Fitch produce credit ratings for issuers of debt instruments. Rating agencies use proprietary models to classify the issuer and the bond issue into one of several discrete credit rating classes. For example, S&P and Moody s use the following classification: 16

17 2.1. Agency Credit Ratings 17 Long Term Obligation Ratings S&P Moody s Meaning AAA Aaa Highest quality, minimal credit risk AA Aa High quality A A Strong payment capacity BBB Baa Adequate protection, moderate credit risk BB Ba Likely to pay, but ongoing uncertainty B B High risk CCC Caa Current vulnerability to default CC Ca C C Nonpayment highly likely; in default (Moody s) D in Default The rating categories between AA/Aa and CCC/Caa can be appended with a + or - sign (S&P) or with 1,2,3 numbers (Moody s) to show relative standing within the category. Obligors rated BBB-/Baa3 or better are called investment-grade ; investors rated BB+/Ba1 or worse are called speculative-grade. Some institutional investors (eg, insurance companies, pension funds, money market funds) are not allowed to hold speculative-grade bonds. Credit rating serves as an independent certification of the ability of the issuer to repay its bonds and loans. It helps the firm to sell its bonds to a large pool of small (otherwise uninformed) investors. Thus, the cost of certification is usually paid for by the issuer. Agencies revise their ratings regularly. They have to strike a balance between rating stability and rating accuracy Transition Matrices and Credit Migration 1-period transition To use agency ratings in a quantitative risk management model, we need to map the letter ratings to credit migration probabilities. In other words, we need to assign numbers to the likelihood that an issuer moves up or down in the credit class or even defaults. Rating agencies also provide this information, conveniently summarized in a transition matrix. For example, Standard and Poor s published the matrix in table 2.1. Each entry represents the probability of migrating from the row-class to the column-class. For example, one year from now a BBB-rated obligor will

18 2.1. Agency Credit Ratings 18 Table 2.1: One-Year Global Corporate Average Transition Rates( , S&P) (%) From/to AAA AA A BBB BB B CCC/C D NR AAA AA A BBB BB B CCC/C have the following rating: BBB with 85.44% probability; BB with 3.75% probability; etc; and will have defaulted with 0.20% probability. Note that each row adds up to 1: next year, the BBB firm must either be in some rating class or it may no longer be rated (NR). Note that the large values in the diagonal reflect the rating stability aimed for by the rating agencies. Example Transition rates are estimated from historical frequencies. Suppose there are only two rating categories: Survival (S) and Default (D). In a portfolio of 300 loans, 60 defaulted during last year. What are the one-year transition rates? S S D N-period transition Given a 1-year transition matrix, we can go even further and compute n- period transition matrices (probabilities of credit migration n-periods from now). First, we need to transform the transition matrix (like the one in table 2.1) into a square matrix. The transition matrix can be made square by: 1. Eliminating the last column ( NR, Not Rated for S&P, WR, Withdrawn Rating for Moody s) by normalizing each transition frequency by the total fraction of bonds that do not have a withdrawn rating.

19 2.1. Agency Credit Ratings Adding one more row in the end for the Default state. This row would have zeros everywhere, except in the last entry which would be 100% (this assumes that default is an absorbing state). Example Suppose there are just two rating classes and the original transition matrix is: Transition Rates (%) A B D NR A B The corresponding square matrix is: A B D A 87/ B D To find the probability of going from state i to state j in 2 periods, one must consider all possible paths that lead from i to j. The probability is thus the sum of all such possible paths, p (2) i j = K k=1 p(1) i k p(1) k j, where K = 8 for a square transition matrix with 8 rating classes (like the one in table 2.1, after being transformed to a square matrix). Hence, the two-period transition matrix is P (2) = P P. In general, Proposition 2.1.1: n-period transition matrix Given a single-period square transition matrix P, the n-period transition matrix, P (n), is the n-th power of P: P (n) = P n Example For the data in example 2.1.1, what is the 2- year transition matrix? First, make the matrix square:

20 2.1. Agency Credit Ratings 20 S D S D Then, P (2) =... = [ ] (Note that the 2-yr default probability is 36% as in example 1.4.1) Assumptions This procedure to extrapolate from 1-period to n-period transition matrices relies on two assumptions: 1. Time-invariance. We assume that the 1-period transition matrix is constant, i.e., does not depend on calendar time. 2. Markov-property. Rating migration probabilities do not depend on anything else besides the current rating. In particular, history (upgrading/downgrading) is irrelevant. These two assumption may be criticized. First, there is evidence that downgrading is more likely in recessions than in boom phases of the business cycle. Second, there is evidence that rating momentum matters, i.e., recently downgraded obligors are more likely to be downgraded again than other obligors whohavebeeninthesameratingclassforalongtime. Nonetheless, thesetwo assumptions are used in practice since they allow considerable simplification of the credit risk models Drawbacks of ratings While investors routinely use ratings to infer probabilities of default, there are several reasons to be careful about this procedure: 1. Even though CRAs publish transition matrices and a lot of other statistics on defaults, they do not say that ratings imply a specific probability of default. In fact, they say exactly the opposite.

21 2.1. Agency Credit Ratings 21 Credit ratings are opinions about credit risk. [...] Credit ratings can also speak to the credit quality of an individual debt issue, such as a corporate or municipal bond, and the relative likelihood that the issue may default. Credit ratings are not absolute measure of default probability. Standard and Poor s internet page Understanding Ratings, The main three CRAs claim to rate through-the-cycle and emphasize that ratingsarejust a relative ranking offirms. This issaying that they will not downgrade everybody just because we are in a recession. 3. The fact that CRAs are paid by issuers generates a conflict of interest. In the wake of the 2008 financial crisis, the US government sued S&P (in Feb/2013) for deliberately inflating ratings and understating the risk associated with mortgage securities (CDOs). In April 2013, S&P responded in court by saying that its own claims to have objective ratings, not corrupted by conflicts of interest, were mere puffery. S&P is basically saying that we should view their claims of independence in the same way that we view a best pizza in the world advertisement. 1 Figure 2.1: Mere puffery 4. CRAs have total discretion in their rating systems and are not required to make their rating methodology public. 5. While some rating agencies aim to reflect cross-sectional variation in default probabilities (like S&P and Fitch), others aim to also incor- 1 In Feb/2015, S&P agreed to pay a 1.5 billion dollar fine to settle the case, without admitting wrongdoing. S&P was the only credit rater sued by the U.S. Justice Department, even though its competitors also issued top ratings for similar subprime-backed securities. Unrelated fact: S&P had downgraded the US s AAA debt rating to AA+ in August 2011.

22 2.2. Credit Scoring and Internal Rating Models 22 porate loss given default and reflect dispersion in expected loss (like Moody s). 6. CRAs claim to try to reach a balance between ratings stability and accuracy. This leads them to revise ratings only between 1 and 2 years on average. 2.2 Credit Scoring and Internal Rating Models Agency ratings are usually only available for large corporations. However, a bank typically has medium and small firms in its credit portfolio. To assess the credit risk of small firms, several statistical methods have been developed. Models of credit scoring may use the following information to forecast the default likelihood: Balance-sheet data measuring the indebtedness of the obligor; Balance-sheet and income-statement data measuring the ability to pay (eg, EBIT and free cash flows); Riskiness of the business (volatility); The firm s market capitalization (if publicly traded); Macroeconomic data Z-Score The first credit scoring model is the Z-score developed by Altman (1968). Using a sample of traded manufacturing companies, Altman proposed the following empirical model.

23 2.2. Credit Scoring and Internal Rating Models 23 Definition 2.2.1: Z-score model (public firms) The Z-score function for a publicly traded firm is Z = 1.2X X X X 4 +X 5 (2.1) where The rule is: X1 := Working Capital / Total Assets X2 := Retained Earnings / Total Assets X3 := EBIT / Total Assets X4 := Market Cap / Total Debt X5 := Sales / Total Assets Z > 1.81: good firm, concede credit Z < 1.81: bad firm, deny credit Example The table below shows the relevant ratios for two firms (The values for Firm Bad are the average values for firms that defaulted within 1 year in Altman s (1968) sample; firm Good has the average values for surviving firms). Variable Firm Bad Firm Good X1-6% 41% X2-62% 35% X3-32% 15% X4 40% 248% X5 150% 190% We can use the Z-score model to determine whether to concede or deny credit. For the first firm, Z bad = 1.2 ( 0.06)+1.4 ( 0.62) = 0.26 < 1.8 Hence, we should deny credit to firm Bad. Z good = = 4.87 > 1.8 Hence, we should concede credit credit to firm Good.

24 2.2. Credit Scoring and Internal Rating Models 24 Statistical details: Recall that there are two possible errors in a loan decision: concede a loan to a bad obligor (type I error); deny credit to a good obligor (type II error). The statistical technique used to determine the cutoff level and the coefficients in (2.1), called linear discriminant analysis, chooses these parameters to maximize the number of correctly classified obligors. Figure 2.2 illustrates a simple case with two variables. Discriminant analysis finds the best line to separate firms that defaulted (X) from firms that survived (O). Figure 2.2: Discriminant Analysis When calibrating the model it is important to construct a sample that includes enough defaulted and surviving obligors. In other words, a data set consisting exclusively of loans conceded by a given bank may lead to biased results (the clients were probably screened for credit quality by some process already in place). Remarks: Thecutoffof1.8proposedbyAltman(1968)isdependentonthesample period and composition. In subsequent studies, Altman advocates a cutoff of Altman, Haldeman, and Narayanan (1977) later proposed a refinement of the model. The model has been sold commercially as ZETA scoring model, but the model is proprietary, so details are not publicly known.

25 2.2. Credit Scoring and Internal Rating Models Z -Score If the firm is not traded, the model proposed by Altman is the Z -score. Definition 2.2.2: Z -score model (private firms) The Z -score function for a private firm is Z = 0.717X X X X X 5 (2.2) where X4 := Book value of Equity / Total Debt X1, X2, X3, X5 are as defined in (2.1) The decision criterion is: Z > 2.90: Safe zone ( good firm, concede credit) 1.23 Z 2.90: Gray zone Z < 1.23: Distress zone ( bad firm, deny credit) Z -Score In a sequence of subsequent papers, Altman further improved the model to allow for private firms (without publicly traded stock), for firms outside the manufacturing sector, and for firms in emerging markets. Definition 2.2.3: Z -score model (any firm) The general Z -score function for any firm is Z = 6.56X X X X 4 (2.3) Z s = Z (2.4) where Z s stands for Standardized Z -score and all X variables are as defined in (2.2). The decision criterion is: Remarks: Z s 4.50 Z s > 5.85: Safe zone ( good firm, concede credit) 5.85: Gray zone s < 4.50: Distress zone ( bad firm, deny credit) Z

26 2.2. Credit Scoring and Internal Rating Models 26 Even though this model does not give directly a probability of default, we can do an empirical match between Z-scores and observed ratings. Then, we can use the probability of default associated with that implied rating. Altman and Hotchkiss (2006) propose the correspondence in table 2.2. For practical applications, the model coefficients and cutoff values should be calibrated to the particular market and industry that we need to analyze, using a recent sample of defaulted and surviving firms. Table 2.2: Correspondence between Z s and Ratings Lower Upper Rating Lower Upper Rating 8.15 > 8.15 AAA BBB AA BB AA BB AA BB A B A B A B BBB CCC BBB CCC CCC < D From Altman and Hotchkiss (2006, fig 12.1) Example Figure 2.3 shows a Bloomberg screen with data for Hellenic Telecommunicatons. 1. Using the data displayed, check that the Z and Z scores are correct. 2. Compute the Z s 3. According to the Z-score and the Z s -score, is the firm Safe or Distressed? 4. What is the rating implied by the Z s-score? (For reference, the actual S&P rating on this date was B+)

27 2.2. Credit Scoring and Internal Rating Models 27 Figure 2.3: Bloomberg screen with Z-scores for HTO Estimating the probability of default The same type of information used in the Z-score model can be used to estimate directly the probability of default. Two commonly used statistical techniques are the probit and the logit models. Model setup Let f := b 0 +b 1 X 1 + +b N X N where X n,n = 1...N, are explanatory variables (similar to the ones discussed in the previous section) and b 0,b 1,...b N are parameters to be estimated. Probit. The probit model estimates the probability of default for a given firm as p = Φ(f) where Φ(.) is the standard normal c.d.f. (thus ensuring a probability between 0 and 1).

28 2.2. Credit Scoring and Internal Rating Models 28 Logit. The logit model is very similar. Instead of a normal distribution, the logit modes uses the logistic distribution for the probability of default: p = ef 1+e f (2.5) Note that 0 < ef 1+e f = 1 1+e f < 1, thus ensuring 0 < p < 1. Estimation The results from the two models are very similar, but the logit model is easier to estimate. The method to estimate the logit model is Maximum Likelihood Estimation (MLE). The Likelihood (L) function is the probability of observing our sample: L = I (p i ) y i (1 p i ) 1 y i i=1 where I is the number of firms in the dataset, p i is the function in (2.5) for firm i, y i = 1 if firm i defaulted and y i = 0 if i did not default. The coefficients b 0...b N are the numbers that maximize the log likelihood function: I ln(l) = y i ln(p i )+(1 y i )ln(1 p i ) i=1 To do this in Excel, use the Solver. In Matlab, use the glmfit() function from the statistics toolbox. Application One important reference for the application of these models is Campbell, Hilscher, and Szilagyi (2008). They find that the following logit model successfully predicts the probability of default for U.S. firms.

29 2.2. Credit Scoring and Internal Rating Models 29 Logit model of Campbell, Hilscher, and Szilagyi (2008) The probability of default over the next month is with p = 1 1+e f Net Income f = Liabilities Assets Assets 7.35 ExcReturn+1.48 Sigma RelativeSize 2.4 Cash Equity Market Share Price Assets Book Equity All these variables are calculated in very specific ways see the paper for details. Remarks: Contrary to the Z -score, this model was calibrated only to the U.S. market and uses a large number of explanatory variables, including some very market-specific variables (like relative size). Therefore, for practical applications, it is very important to fully reestimate the whole model with a relevant sample for the particular market that we need to analyze. It is even likely that different variables will become significant Application: loan pricing Once we have a logit model calibrated, we can use it to price new loans. First, we estimate the probability of default for the new borrower through (2.5). Then the spread is determined as follows. Banks typically measure their performance through risk-adjusted measures such as the Risk-Adjusted Return on Capital (RAROC): where RAROC = R C EL K

30 2.2. Credit Scoring and Internal Rating Models 30 R is the revenue. For a loan of F dollars and spread s, R = (LIBOR+ s) F. C are the costs. For simplicity consider only funding costs: C = LIBOR F. In general, one should also consider administrative and operational costs. EL is the expected loss, EL = PD F LGD, assuming that the exposure at default is F. K is the capital at risk or economic capital. Assume this is a fraction of the loan principal, K = kf, with k = 8% (Basel I regulatory capital value for corporate loans). 2 The loan should be made if RAROC ROE where ROE is the Return on Equity required by shareholders. Solving this equation for s, we find that the minimum spread is s = ROE k +PD LGD See the course s website for an application. 2 In Basel II this was modified to K = 0.08 RWA, with the Risk-Weighted Assets depending on the risk of the specific asset or borrower through more elaborated procedures. The current Basel III further increases the required regulatory capital.

31 Chapter 3 Structural approach to credit risk The structural approach relates credit risk to fundamental variables, such as the value of the assets of the firm. The seminal model is Merton (1974). This model is the reference framework for some of the most important commercial models of credit risk management (e.g., CreditMetrics, and Moody s/kmv EDF). 3.1 Merton s model Assumptions Capital Structure In Merton s model the firm has a simple capital structure: (1) equity, S t ; (2) a single zero-coupon debt instrument maturing at time T, with face value F, and current market value B t. The value of the assets is V t. Hence, the market-value balance identity is V t = S t +B t. At maturity of the debt, if V T < F then it is rational for the shareholders to default onthe debt (thereis limitedliability, so theyjust need to surrender the assets to creditors). Hence, there is credit risk as long as P[V T < F] > 0. Figure 3.1 illustrates the default region in the distribution of V. 31

32 3.1. Merton s model 32 Figure 3.1: Default Region Distribution of Asset Value To compute the exact probability of default we must specify the distribution of assets value. The assumption in Merton s model is that V t follows a geometric Brownian motion dv t /V t = µdt+σdw t where W t is a standard Brownian motion 1, and µ and σ 2 are the mean and variance of the instantaneous rate of return on the assets (dv t /V t ). It can be shown that the value of V at any future time t is given by } V t = V 0 exp {(µ )t+σw σ2 t (3.1) 2 with W t N(0,t). 2 1 W 0 = 0, W t N(0,t), independent increments. 2 By definiton, Y is lognormally distributed if lny = X N(m,s 2 ). The mean is given by E[Y] = E[e X ] = e (m+1 2 s2 ). Thus, the equation basically means that Vt follows a lognormal distribution with mean E[V t ] = V 0 exp(µt).

33 3.1. Merton s model 33 V t t Therefore, the continuously compounded assets return (R : V t = V 0 e R ) is normally distributed: R ln V ) ) t N ((µ σ2 t,σ 2 t V 0 2 The normalized assets return r follows a standard normal distribution: ( ) ln Vt V 0 µ σ2 t 2 r σ N(0,1) (3.2) t Probability of Default Let p Def denote the probability of default at the horizon T. Given the assumptions above, it is p Def = P[V T < F] [ = P ln V T < ln F ] V 0 V 0 ( ln F V 0 = P r < σ T µ σ2 2 ) T r Def = N(r Def ) (3.3) where N(.) is the standard normal c.d.f. Hence, we just have to read the probability mass to the left of the critical value r Def from the table of the standard normal distribution N(0, 1).

34 3.1. Merton s model 34 Example A firm has debt with face value F = 70 and its assets are currently worth V 0 = 100. The assets return process has been estimated to have an instantaneous mean of 10% per year, µ = 0.1, and standard-deviation of 20% per year, σ = 0.2. Check that the probability of the firm defaulting in one year is: p Def =... = Price of a corporate bond The fundamental insight of Merton is to realize that holding a defaultable bond is the same as holding a risk-free bond plus being short a put option on the value of the firm s assets: defaultable bond = long riskless bond + short put Figure 3.2 shows the payoffs at maturity. Formally, B T = min(f,v T ) = F max(0,f V T ). Figure 3.2: Merton s Insight F Value of Debt = Riskless Debt + Short Put Payoff Long Put 0 F Value of Assets To see this from a different perspective, consider a bank loan instead of a bond. Suppose the bank conceding the loan also buys an european put

35 3.1. Merton s model 35 option on V, with maturity T, and strike price F. The cash flows to bank are the following: t = 0 t = T Value of assets V 0 V T < F V T F State Default Survival Bank s position 1) Loan to firm B 0 V T F 2) Long put P 0 F V T 0 Total B 0 P 0 F F By spending P 0 to buy the put option, the bank converts the risky corporate loan into a riskless loan with face value F. Hence, credit risk can be eliminated by buying a put option on the assets of the firm. In other words, P 0 is the cost of eliminating the credit risk. Bond Price. In equilibrium, this strategy must produce a return equal to the risk-free rate (i), i.e., or B 0 +P 0 = Fe it B 0 = Fe it P 0 (3.4) (again, risky debt equals riskless debt plus short put). Put price. Using the Black-Scholes option pricing formula, we get the value of this put option: P 0 = Fe it N( d 2 ) V 0 N( d 1 ) (3.5) where i is the risk-free interest rate, N(.) is the standard normal c.d.f., and 3 d 1 = ln(v 0/F)+(i+σ 2 /2)T σ T d 2 = ln(v 0/F)+(i σ 2 /2)T σ T = ln V 0 Fe it +σ 2 T/2 σ T (3.6) = d 1 σ T (3.7) 3 Note the similarity between d 2 and r Def. However, in r Def we worked with the actual return distribution (thus the assets grow at rate µ), whereas here we are using the risk-neutral option pricing framework (d 2 is a function of the risk-free rate).

36 3.1. Merton s model 36 The ratio of the present value of the debt obligation (assuming the debt is riskless) to the current value of the assets, L Fe it /V 0, is called the Leverage Ratio. Example Continuing the previous example with assets V 0 = $100 and σ = 0.2, and debt F = $70 maturing in one year (T = 1), further assume that the continuously compounded risk free interest rate is i = 5%. First, compute the put price: d 1 =... = d 2 = = P 0 =... = $ Second, compute the bond price: B 0 =... = $66.46 To verify that if the bank buys the put, it will indeed get a return equal to the risk-free rate, we solve B 0 +P 0 = Fe x = 70e x x = 5% Call price. Further note that Equity can be seen as a call option on the assets of the firm (with strike price equal to the value of debt and same maturity). Hence, we can use the Black-Scholes call option pricing formula: S 0 = V 0 N(d 1 ) Fe it N(d 2 ) (3.8) Example Check that in the previous example the firm s equity is worth: S 0 =... = $33.54

37 3.1. Merton s model Default Spread Let y denote the yield to maturity on the risky debt: B 0 = Fe y.t y T = 1 T ln B 0 F Since there is default risk, theytm must begreater thanthe risk-free rate, y > i. The default spread is defined as the difference between these rates: s y i We can also compute the spread directly from the parameters of the model. Replacing (3.4) for B 0, we have y = 1 ln Fe it P 0. Then, using the T F spread definition and replacing (3.5) for P 0, s = y i = 1 T ln Fe it P 0 F = 1 ( T ln 1 P 0 Fe it 1 T ( 1)lne it ) = 1 T ln ( 1 N( d 2 )+ V 0 ) Fe itn( d 1) ) = 1 T ln ( N(d 2 )+ V 0 Fe itn( d 1) Example Continuing the previous example, the ytm on the firm s debt is: y 1 T ln B 0 F =... which means that the spread is s =... = 0.19% Alternatively, we can compute the spread directly by s = 1 ) (N(1.9334)+ 1 ln e N( ) = See Lando (2004) for a survey of extensions of Merton s model.

38 3.2. Moody s-kmv EDF Moody s-kmv EDF Overview The consulting firm KMV developed a very nice practical implementation of Merton (1974) model. KMV s model is structural in the sense that links default probabilities to fundamental variables. 4 KMV s initial motivation was the fact that agency credit ratings typically change slowly, in discrete updates, while default probabilities change continuously (KMV showed in a study that some bonds rated BBB and AA may in fact have the same probability of default). The KMV approach can be applied even in countries where there is not enough historical information to estimate transition matrices. However, it is best applied to publicly traded companies, where equity prices can be observed in the market. 5 Procedure The main output of KMV s model is the so called Expected Default Frequency (EDF), which corresponds to the default probability of a given obligor. It is determined in three steps: 1. Estimate the assets value and volatility; 2. Calculate the distance to default ; 3. Map the distance to default to the probability of default (EDF) Estimation of the assets process If all liabilities were traded in the market, we could measure the value of the firm s assets by simply adding up the value of equity and debt. However, in practice not all debt is traded, so we cannot directly observe asset market values. 4 KMV was acquired by Moody s in 2002 and became Moody s KMV. 5 Nonetheless, there is also a Private Firm Model which, in the absence of equity market values, estimates the assets value and volatility from accounting data and comparable companies.

39 3.2. Moody s-kmv EDF 39 Intuition: simple capital structure We could find the two unknowns V 0 and σ v by solving the following two equations. In the Black-Scholes-Merton framework equity itself can be seen as a call option on the assets of the firm (with strike price equal to the value of debt and same maturity). Hence, we can use the Black-Scholes call option pricing formula (eqn 3.8): S = V 0 N(d 1 ) Fe it N(d 2 ) Furthermore, the volatility of the equity (σ s ) is related to the volatility of the underlying (assets in this case, σ v ) through 6 σ s = N(d 1 ) V 0 S σ v Given that equity is observable, we can use these two equations to back out the value of assets V 0 and its volatility σ v. Still, the solution is not trivial since d 1 and d 2 depend on σ v and V 0. More realistic capital structure KMV extends the Black-Scholes-Merton (BSM) model to allow for a more realistic capital structure. In the current KMV model, the firm may have 5 classes of liabilities: short-term debt, long-term debt (in perpetuity), convertible securities (preferred stock or bonds), preferred stock, and common stock. We now get more generic functions relating the variables of interest: S = f(v,σ v,i,capital structure) σ s = g(v,σ v,i,capital structure) Furthermore, the equations are different from the simple BSM model because KMV considers equity as a barrier, perpetual call option, consistent with the firm being an ongoing concern (otherwise the inferred V 0 and σ v would be a function of time-to-maturity, making them meaningless). 6 The first equation says S = f(v,t;t,f,i,σ v ). Applying Ito s lemma, ds = f t dt + f V dv f VV(dV) 2 where dv t /V t = µdt +σ v dw t. Hence, ds = (f t + µvf V σ2 v V 2 V f VV )dt+σ v Vf V dw. Dividing by S, ds/s = (...)/Sdt+σ v S f V dw. The volatility of the stock is the term in front of dw: σ S = σ v S f V. Using the fact that the delta of V V the option is f V = N(d1), we get σ S = σ v S N(d1)

40 3.2. Moody s-kmv EDF 40 Estimation of µ We also need to estimate the expected growth rate of the assets, µ. However, expected returns are notoriously hard to estimate. One possibility is to use a unique µ per sector or industry (which may be less subject to estimation error). Another possibility is to use one of the alternative estimation procedures described below Distance to Default Default point In reality, when firms have several different types of debt, default is not so straightforward as in the idealized Merton framework (default when V < F). Using a large sample of firms, KMV observed that default happens when the asset value reaches a level somewhere in between total liabilities and short-term debt. Instead of using the total value of debt, KMV defines a critical threshold for default called Default Point (D* or DPT) as the sum of the par value of current short-term debt (STD) with half of the par value of current long-term debt (LTD): D := STD +1/2 LTD DD: Intuition KMV then computes an intermediate index called Distance to Default (DD), which measures how far the firm is from D : DD := V 0 D σv 0 It measures the distance to default in terms of the standard-deviation of the assets. DD: Current KMV model In the current model, the DD formula is made more precise assuming a lognormal distribution for V, as in Merton. Recall equation (3.1), which implies ) lnv t = lnv 0 + (µ σ2 t+σw t (3.9) 2

41 3.2. Moody s-kmv EDF 41 with ( W t ) N(0,t). Hence, for some horizon T, we have E[lnV T ] = lnv 0 + µ σ2 T and Var[lnV 2 T ] = σ 2 T. KMV now computes the Distance to Default as ( ) lnv 0 + µ σ2 T Payouts lnd 2 DD := σ T (3.10) where Payouts reflect the asset drainage through cash outflows until T (debt coupons and preferred and common dividends). Note that this is estimating the distance to default at the horizon T, not at the current time 0 as in the basic initial model Probability of Default (EDF) Under the Gaussian distribution of the Merton model, we would have PD = N( DD) To see this, note that the DD defined in (3.10), ignoring payouts, is just the symmetric of the critical value defined in (3.3). However, the PDs computed with a normal distribution are typically unreasonable. 7 KMV uses instead a large sample of historical information to make an empirical mapping from DD to actual default probabilities. KMV calls these estimated probabilities Expected Default Frequencies (EDF). This proprietary database is not publicly available, so we cannot replicate this last step of the procedure. Nevertheless, the intuition is simple. Suppose that among a population of 5000 firms with DD = 4 at one point in time, 10 defaulted one year later. Then, the expected default frequency would be EDF = 10/5000 = 0.2%. Any firm with DD close to 4 will be assigned an EDF of 0.2%. Naturally, the EDF decreases with DD. Example Consider the following example provided by KMV (note that it uses the basic old definition of DD). It measures the company Federal Express on two different dates. 7 DPs for firms with high DD would be absurdly low because the tails of the Normal distribution are not thick enough.

42 3.2. Moody s-kmv EDF 42 Nov 97 Feb 98 Market cap $ 7.7 $ 7.3 Book liabilities $ 4.7 $ 4.9 Market value of assets $ 12.6 $ 12.2 Asset volatility 15% 17% Default Point $ 3.4 $ DD EDF 0.06% 0.11% The increase in the EDF is explained by the decrease in the stock price, increase in debt, and increase in asset volatility (business more risky) Summary of Differences to Merton The KMV EDF model was developed by Vasicek andkealhofer (the V and K in KMV). As discussed above, they extended Black-Scholes-Merton s framework in the following dimensions: Black-Scholes-Merton 2 Classes of Liabilities: Debt and Equity No Cash Payouts Default occurs only at Horizon Default barrier is total debt Equity is a call option on assets, expiring at maturity of debt Gaussian relationship between DD and probability of default Vasicek-Kealhofer EDF model 5 Classes of Liabilities: Short-term and long-term debt, convertible securities, preferred stock, and common stock. Cash Payouts: Coupons and Dividends Default can occur at or before Horizon Default barrier is empirically determined Equity is a perpetual call option on assets Empirical mapping from DD to EDF from historical data EDF versus Agency Ratings KMV s approach has some advantages relative to agency (S&P, Moody s, etc) ratings:

43 3.2. Moody s-kmv EDF EDF usually anticipate (sometimes by 1 year or more) the downgrading of the issuer by rating agencies. Figure 3.3 shows Enron s 2001 bankruptcy case. KMV shows similar graphics for other important bankruptcies, such as Worldcom (America s largest bankrupcty, 2002) and United Airlines (2002). 2. EDF follows the business cycle: increases in recessions and decreases in good times. Basically, these advantages come from the fact that KMV uses equity data to estimate the model parameters, and is thus able to use information revealed in the stock market. On the other hand, since agency ratings are attributed with stability in mind, they tend to be better long term predictors of credit quality (at least under normal circumstances) Alternative estimation procedures Iterative procedure Instead of the procedure described above of solving two equations for two unknowns, we can instead use the following iterative procedure on only one equation: 1. Use one year of daily stock returns to compute the volatility of equity and use this as a starting value for the volatility of assets, σ v. 2. Solve (3.8) at each day, assuming that F equals total debt and that all debt is due in 1 year from that day. This results in a daily time series for V t. 3. Use this series to obtain the next estimate of σ v. 4. Go back to step 2 and repeat this procedure until the estimates of σ v converge, that is, until the distance between two consecutive estimates is less than, say, The final time series of V t is used to estimate µ. 8 Note however that there is little independent verification of these advantages; these are KMV s or Moody s/s&p versions of the story.

44 3.2. Moody s-kmv EDF 44 Figure 3.3: Enron Default

45 3.2. Moody s-kmv EDF 45 This procedure has been used in several papers (eg, Vassalou and Xing (2004)) and Moody s-kmv seems to be following a somewhat similar version of this procedure (see Dwyer and Qu, 2007). Ad-hoc variables, but preserving functional form Bharath and Shumway (2008) propose the following simpler procedure to estimate the probability of default. Alternative estimation of PD Bharath and Shumway (2008) propose to estimate the probability of default by: ln ( ) ( S+F F + r s σ2 v 2 )T PD = N( DD), with DD = σ v T where F = face value of total debt S = market capitalization σ v = S S+F σ s + F S+F σ d, with σ d = σ s σ s = annualized stock volatility estimated over the previous year r s is the stock return over the last year (proxy for µ) T is the horizon (typically T = 1 year) Example Use the data for HTO in figure 2.3. Using daily stock returns over the previous year, we obtain r s = 4.26% and σ s = 60.66%. 1. Check that PD =... = 10.58% 2. Is the signal coming from this PD consistent with the signal from the Z s -score model in exercise 2.2.2?

46 3.2. Moody s-kmv EDF 46 Remarks: 1. Bharath and Shumway (2008) show that their simple probabilities correlate well with (a small sample of) Moody s-kmv EDF. 2. They further show that their simple probabilities are useful to predict defaults, in the sense that their PD shows up as a significant variable to explain defaults in an econometric reduced-form model. 3. This means that their probability is useful to rank firms by their likelihood of default, even if the actual probability value may not be the true probability of default.

47 Chapter 4 Portfolio Models This section presents several models of portfolio credit risk. Banks are have to pay close attention to the credit risk of their overall loans portfolio, as the stability of the bank depends to a large extent on the performance of the portfolio. Credit event correlations are the key factors driving the risk of the portfolio. However, estimating correlations is a hard task due to data scarcity on defaults. We will therefore need simplifying assumptions to be able to estimate risk measures. The CreditMetrics model presented in this section offers an ingenuous solution to this problem based on an extension of Merton s model. An important empirical fact about credit returns (returns on a portfolio of loans or bonds) is that they are not normally distributed, in contrast to market returns (see figure 4.1). The skewness in the distribution is caused by credit losses. Hence, the mean and variance will not be sufficient statistics to understand the distribution of credit returns. We will need to estimate Credit VaR directly from the full distribution of credit returns. 4.1 Credit Migration Approach CreditMetrics The Credit Migration method estimates future changes in the value of a portfolio of loans and bonds that are due to changes in the credit rating of the obligors. The method is based on the standard rating categories and respective transition matrices (see section 2.1). 47

48 4.1. Credit Migration Approach CreditMetrics 48 Figure 4.1: Portfolio Returns Assumptions The industry implementation of this method is done by J.P.Morgan, as CreditMetrics. CreditMetrics does the following simplifying assumptions: 1. The only source of uncertainty is credit migration, i.e., moving up or down in the rating scale. 2. Within the same rating class, all issuers are homogeneous. That is, they have the same transition and default probabilities. 3. Interest rates are deterministic (as if there was no uncertainty). Hence, future bond values can be computed using today s forward rates. 4. Equity value is a good proxy for the value of the assets of the firm. Hence, correlations between asset values can be approximated by correlations between stock prices. (In reality, equity is more volatile than assets due to leverage). The CreditMetrics method uses two main building blocks: 1. Credit risk for a single instrument. 2. Credit risk for the whole portfolio, accounting for correlations between credit events.

49 4.1. Credit Migration Approach CreditMetrics 49 The next two sections discuss these two blocks using the examples in the Credit Metrics Technical Document (J.P. Morgan, 1997) Single exposure risk To estimate thechange invalueofasinglebondover agiven horizon(1year), we just need to compute the bond price for each possible credit migration. We can then use this distribution to compute measures of risk. Suppose we hold a 5-year, 6% annual coupon, BBB-rated bond. To estimate its credit risk we proceed in the following steps: 1. Estimate/obtain a transition matrix. The transition matrix is in table 4.1 Table 4.1: Transition Matrix 2. Estimate the recovery rate for the seniority characteristics of the bond. This allows the valuation in the state of default. Our BBB bond is senior unsecured and we estimate its mean recovery rate to be 51.13% (recall the empirical studies in section 1.3). 1 1 The model can also accommodate uncertain recovery rates. See J.P. Morgan (1997) for details.

50 4.1. Credit Migration Approach CreditMetrics Use forward interest rates for each rating category to value the bond in each state. Suppose we estimate the term structures in table 4.2 (note that there is one term structure for each rating class) Table 4.2: Interest rates for each credit rating We can compute the value of the bond one year from now in each possible state by discounting the cash flows at the relevant rates. For example, if the bond upgrades to single-a rating, its value will be V A = = Proceeding in the same way for other rating classes, we get the estimates in table 4.3. Note that the table actually shows for this case (instead of ). This is probably due to rounding errors in the interest rates. 4. Compute credit risk measures. We now have all information to construct the distribution of value changes. Table 4.4 shows the results. Columns two and three repeat the probability and value in each state. The expected value of the bond is the average across all states: µ V s p s V s = $107.09

51 4.1. Credit Migration Approach CreditMetrics 51 Table 4.3: Possible Bond Prices Table 4.4: Changes in Bond Prices

52 4.1. Credit Migration Approach CreditMetrics 52 The change in value in each state is the difference to the mean, V s V s µ V. For example, V A = = $1.57 which will happen with probability p A = The fifth column shows all possible values. From the distribution of V we can compute the desired loss percentile. For instance, to find the maximum loss at the 99% confidence level, start accumulating the probability from the left and stop when you reach 1%. For this example, the critical value is between ( CCC s=d p s = 0.3%) and ( B s=d p s = 1.47%). This calculation is better suited for a portfolio, where the distribution of V will be more continuous. For now, we can adopt a conservative estimate and state that credit risk of our BBB-rated bond is Credit VaR 1yr 99% = $23.45 Or, we can interpolate between the two values, to get (Interpolated) Credit VaR 1yr 99% = $14.80 = VaR, Note that CreditMetrics considers losses from downgrades, not only from default. An alternative measure of risk also computed by CreditMetrics is the standard deviation of the bond value: σ 2 V s p s (V s µ V ) 2 = s p s ( V s ) 2 = and σ V = $2.99 Note that assuming a normal distribution for V would lead to a much different credit VaR. For this example, given that the standard normal critical value is z 99% = 2.326, we would get Credit VaR 1yr 99% = µ V V 1% = µ V (µ V z 99% σ V ) = ( ) = $6.95. This value substantially underestimates the credit risk of the bond. The reason is that the distribution of credit losses, and thus of V, is not normal. 2 2 Note that V and V V µ V havesimilardistributions; only the support (horizontal axis) changes.

53 4.1. Credit Migration Approach CreditMetrics Portfolio risk Suppose we have an additional bond in the portfolio, i.e., we now have the following bonds: 1. BBB rated, senior unsecured, 6% annual coupon, 5-year maturity; 2. A rated, senior unsecured, 5% annual coupon, 3-year maturity; Portfolio values Proceeding in the same way as before, we can determine the distribution of possible year-end values for bond A. Considering the major 7 rating categories plus the default state, each bond can be in one of 8 states at the end of the year. Thus, the portfolio can be in 1 of 64 states. The value of the portfolio will be the sum of the individual values. Table 4.5 displays all possible portfolio values. Table 4.5: Portfolio Values Example Check the value in cell A,A.

54 4.1. Credit Migration Approach CreditMetrics 54 To compute the Credit VaR we need the likelihood of each of these 64 states. Determining the joint migration probabilities becomes the crucial task. Joint probabilities with independent migrations Suppose that the two bonds are independent of each other. In this case, the joint likelihood is simply the product of the individual likelihoods. For example, using the transition matrix in table 4.1, the probability that both bonds become BBB-rated in one year is p(bbb BBB,A BBB) = p(bbb BBB) p(a BBB) = = With independence, we could thus easily construct the distribution matrix in table 4.6. Note the following: (1) the most likely outcome is that both obligors remain at the current rating; (2) all probabilities in the table add up to one; (3) the sum along each row (or column) is equal to the chance of migration for that obligor standing alone (for example, the sum of the last row is 0.18%, which is the chance of the BBB obligor defaulting). Table 4.6: Joint Probabilities with Zero Correlation

55 4.1. Credit Migration Approach CreditMetrics 55 Joint probabilities with correlated migrations However, assuming independence is too simplistic because credit migrations are affected in part by the same macro-economic factors. Hence, we need to estimate the correlations between joint migrations. The lack of data on joint credit migration makes it difficult to estimate migration probabilities directly from historical data. For example, since most firms have not defaulted, the observed default correlation would be zero. Therefore, CreditMetric extends Merton s model to be able to link joint migration probabilities to asset values. CreditMetrics generalizes Merton s model by stating that, in addition to the default threshold, there are also credit rating up(down)grade thresholds as well. The firm s asset value relative to these thresholds determines its future rating, as illustrated in figure 4.2. Figure 4.2: Credit Migration Thresholds The relevant thresholds are defined so that the probabilities in the empirical transition matrix match the probability masses from Merton s model. The thresholds are defined in terms of the normalized asset return, as defined in equation (3.2). To calibrate the default threshold r Def for the BBB-rated

56 4.1. Credit Migration Approach CreditMetrics 56 bond, we use equation (3.3) to get p Def = N(r Def ) = N(r Def ) r Def = 2.91 The next threshold, r CCC, can be computed as p Def +p CCC = N(r CCC ) = N(r CCC ) r CCC = 2.75 which means that if the normalized 1-year return on the firms assets falls between and -2.75, our BBB-rated bond will be downgraded to CCC. We proceed in the same way for all thresholds, until the last one p Def +p CCC + +p AA = N(r AA ) = N(r AA ) r AA = 3.54 If r > 3.54, the bond will be upgraded to AAA. We can do the same for our other A-rated bond, and compute its credit migration thresholds. Table 4.7 presents the results. Table 4.7: Normalized return thresholds A-rated obligor BBB-rated obligor Rating in 1 year (s) Prob (%) Threshold (r s (A) ) Prob (%) Threshold (r s (BBB) ) AAA AA A BBB BB B CCC Default The next step is to estimate the correlation between the two firms s asset returns. Since asset returns are not directly observable, CreditMetrics uses

57 4.1. Credit Migration Approach CreditMetrics 57 equity returns as their proxy. Note that this is a drawback of the approach, especially when applied to highly leveraged firms, whose equity returns are more volatile than assets s returns. Nonetheless, it is better than assuming, say, a constant correlation across all assets. 3 Assume that the correlation between the assets of our two firms is ρ = CreditMetrics further assumes that the assets normalized returns are bivariate normal distributed. We can thus use the joint normal distribution to estimate the likelihood of all 64 states in the joint transition matrix. For example, the likelihood that both bonds retain the current rating is = P[ 1.51 < r (A) < 1.98, 1.49 < r (BBB) < 1.53] = f(r (A),r (BBB) ;ρ = 0.3)dr (BBB) dr (A) where f(.) is the joint standard normal distribution. 5 Table 4.8 shows all 64 joint probabilities. Note that the effect of correlation is to increase the mass along the diagonal that goes through the current rating (cell with 79.69). That is, with positive correlation it is more likely that both obligors retain the current rating, are both upgraded, or are both downgraded. Credit VaR of the Portfolio With the information in tables 4.5 and 4.8 we can compute the portfolio VaR in exactly the same way as we did for a single obligor in section First we compute the mean portfolio value, µ V s p sv s = $ Second, we estimate the 1% critical value V (or any other confidence level). Todothis, sorttheportfoliovaluesintable4.5fromthesmallest (biggest loss) to the largest (biggest gain). Start accumulating the 3 The software actually allows the user to input a different correlation, in case he has a better guess. 4 According to J.P. Morgan (1997), the typical asset correlation is in the range of 20% to 35%. 5 If x and y are standard normal with correlation ρ, the joint distribution is { 1 x 2 f(x,y) = 2π 1 ρ exp +y 2 } 2ρxy 2 2(1 ρ 2 ) See Greene (2008, ch 17.3) for numerical techniques for computing multivariate normal probabilities. In Matlab use the function mvncdf() from the statistics toolbox.

58 4.1. Credit Migration Approach CreditMetrics 58 Table 4.8: Joint Probabilities with 0.3 Correlation respective probabilites from table 4.8 and stop when it reaches 1%. The corresponding portfolio value is V = $ The VaR is the distance between the two values: Credit VaR 1yr 99% = µ V V = = $9.23 Remark. This example form Credit Metrics starts with bond prices in percentage of face value, but here we already have a VaR in dollars. This implicitly assumes that the portfolio was initially formed by investing $100 in each bond and that they were both priced at par. Under more general conditions, we would have to compute the terminal dollar value in each bond i through V1 i = V 0 i P i P0 i 1, where V 0 i is the actual amount of money initially invested in the bond and P i is the price (usual percentage of face value). Then, we would sum across all bonds to get the portfolio value, V = i V 1. i Large portfolios For portfolios with many obligors, the analytic approach develop above is not practical. For example, for a portfolio with N = 100 obligors, we would

59 4.1. Credit Migration Approach CreditMetrics 59 have to estimate 4950 pairwise correlations (N(N 1)/2) plus 100 individual variances. CreditMetrics does the following simplifying steps: 1. Factor Model. Pairwise correlations are imposed through an index or factor model. Each firm s standardized return (r n ) is mapped to a set of K << N industry and country factors: r n = K w nk F k +ǫ n k=1 where F k is the factor standardized return, w nk is the loading of asset n on factor k (to be estimated), and the residuals ǫ n are assumed to be independent across assets. Correlations are only estimated between the K factors. This makes it feasible to construct a correlation matrix for a large portfolio. For example, suppose that the N = 100 obligors can be explained by the returns on K = 5 industries. Then, we only need to estimate a 5-by-5 correlation matrix for the factors (15 different parameters), plus 500 loadings, plus 100 residual variances, totalling 615 parameters much less than the 5050 parameters above. Example This example shows how the factor structure imposes correlation between the assets. Once we have the loadings and correlations between the factors, we can easily compute the correlation between any pair of assets. 6 Vivendi Universal is a company involved in telecommunications (T) and leisure (L). Its return is estimated as: r Viv = 0.4T +0.2L+ǫ V iv General Motors is a car manufacturer (industry C): r GM = 0.9C +ǫ GM The factor correlation matrix, estimated from equity indexes, is the following: Correlations T L C T L C This is the example in Servigny and Renault (2004, p.254).

60 4.2. Other models 60 The implied correlation between the two firms is then: 7 ρ Corr(r GM,r Viv ) = Cov(r GM,r Viv ), since var(ret)=1 = Cov(T,C) Cov(L,C) = , since var(factors)=1 = Simulations. Monte Carlo simulation is used to generate the distribution of the portfolio value. This consists of generating a lot of possible scenarios, say 100,000. Each scenario is built by simulating N asset returns (one for each obligor) and computing the resulting portfolio value. The credit VaR is found from this simulated distribution. See J.P. Morgan (1997, ch 8) for details. 4.2 Other models CreditPortfolioView As described above, CreditMetrics uses a constant transition matrix. The matrix is unconditional in the sense that it is an historical average based on many years of data covering several business cycles. However, it is known that default and credit migration probabilities are linked to the state of the economy, with credit cycles following business cycles closely. In recessions, both defaults and downgrades probabilities increase; in expansions, the opposite is true. Wilson (1997a,b) proposes an improvement to the credit migration approach called CreditPortfolioView, which is now commercially implemented by McKinsey. In this method, transition probabilities vary over time. More precisely, they are conditional on the state of the economy, consistent with empirical evidence. CreditPortfolioView uses an index to describe the state of the economy. The index is a function of macro-economic variables such as: - unemployment rate 7 Recall Cov(aX,bY) = abcov(x,y) and Cov(aX,bY + cz) = Cov(aX,bY) + Cov(aX, cz)

61 4.2. Other models 61 - rate of growth in GDP - interest rates - foreign exchange rates - government expenditures - aggregate savings rate Conditional transition probabilities are then estimated as functions of this macro-index. The limitation of this approach is that, being conditional, it requires a larger amount of default and migration data to calibrate the model Moody s-kmv Portfolio Manager Moody s-kmv also sells a model to manage portfolios of obligors. It differs from CreditMetrics in that it does not use credit ratings and thus avoids having to assume that firms are homogenous within a credit rating category. The risk of each individual firm is measured directly by Moody s-kmv proprietary EDF. Furthermore, since the model takes as input the EDF for each obligor, it is able to provide more timely warnings than credit ratings. A factor model is also used to generate correlated asset returns. It outputs the Credit VaR of the portfolio and other relevant risk measures. The technical details of the model are discussed in Smithson (2003, p.162) Reduced-Form Approach CreditRisk+ The Reduced-Form model is also called Actuarial Science model. It is not based on any structural model. There is no economic story to explain why default happens. The default event is exogenous; in a sense, default takes bond-holders by surprise. The actuarial model discussed here is CreditRisk+, implemented by Credit Suisse. Only default risk is modeled; downgrade risk is ignored. It is assumed that: 1. For a single obligor, the probability of default in a given period is the same as in any other period (the default intensity is constant through time). 2. For a single obligor, the probability of default is small. 3. For a large portfolio of obligors, the number of defaults in any given period is independent of the number of defaults in any other period.

62 4.3. Conclusion 62 Under these assumptions, the probability distribution for the number of defaults in a given period of time can be approximated by a Poisson distribution: 8 P[n defaults] = µn e µ, for n = 0,1,2,... n! where µ is the average number of defaults per time period (usually 1 year). The number of defaults per year, n, is itself a random variable, with E[n] = Var[n] = µ. Example Suppose the average number of defaults per year is µ = 3. The probability of observing no defaults in the next year is P[0 defaults] = 30 e 3 = 4.98% 0! The probability of having three defaults is P[3 defaults] = 33 e 3 3! = 22.4% CreditRisk+ then uses this basic statistical model, with some refinements, to derive closed-form expressions for the loss distribution of a portfolio. Credit VaR is then computed in the usual way. CreditRisk+ only focus on default, which has advantages and disadvantages. The main advantage is that it is easy to implement and computationally attractive. The main disadvantage is that it ignores migration risk (downgrading). 4.3 Conclusion Table 4.9 compares the credit portfolio models discussed in this section. All these models may appear to be very different. However, several comparative studies have shown that when these models are run using consistent parameters, the results are quite similar. 8 The Poisson distribution is the limit case of a binomial distribution when the number of obligors in the portfolio goes to.

63 4.3. Conclusion 63 Table 4.9: Comparison of Industry Models CreditMetrics KMV Portfolio CreditRisk+ Manager Model Class Structural Structural Reduced Form or Actuarial Risk source Market value Default loss Default loss Obligor s risk Ratings EDF Exposure bands classification Credit events Downgrade, Default Continuous Default Probability Default Correlation of Factor model Factor model Factor model credit events Recovery rates Random (beta dist) Random (beta dist) Constant Solution Simulation (can be Simulation (can be Analytic (fast) Method slow for large portfolios) slow for large portfolios) % banks using 20% 69% 0% model in Smithson (2003) survey

64 4.3. Conclusion 64

65 Chapter 5 Valuing defaultable bonds We want to value a defaultable bond paying a coupon rate c at dates t i,i = 1,...,m and a face value of 1 at maturity date t m = T. Let C i denote the total cash flow paid at time t i as a fraction of par value (ie, C m includes the notional). 5.1 Credit spreads Alternative spreads The credit spread on a bond is a measure of the additional yield relative to some reference rate. There are several alternative definitions. Definition 5.1.1: G-spread The G-spread for a corporate bond is: where YTM C = YTM G +G-spread YTM C is the yield to maturity on the corporate bond YTM G is the yield to maturity on a Government bond with the same maturity A better alternative for the reference risk-free rate is the Interest Rate Swap curve: 65

66 5.1. Credit spreads 66 Definition 5.1.2: I-spread The I-spread for a corporate bond is: where YTM C = IRS+I-spread YTM C is the yield to maturity on the corporate bond IRS is the swap rate for the maturity of the bond, interpolated from the par Interest Rate Swap curve. Example Bond C is a 3-year bond with 10% annual coupons tradingat102.53%. TocomputeitsI-spreadwefirst needitsytm. Check that ytm = 9%. Second, we need IRS rates: IRS market t market quoted par swap rate (*) 5.00% 5.97% 6.91% (*) Assume all swaps have fixed and floating legs with annual payments. Therefore, I-spread =...= 2.09% A more precise measure uses Z ero-coupon spot rates: Definition 5.1.3: Z-spread The Z-spreadfor a corporatebondis theconstant number z thatsolves V 0 = m i=1 C i [1+r(0,t i )+z] t i (5.1) where V 0 is the price of the corporate bond r(0,t i ) is the Zero-coupon LIBOR-Swap rate for maturity t i (derived from market LIBOR and IRS quotes).

67 5.1. Credit spreads 67 Example Continuing the previous example, first check that the implied zero-coupon IRS rates are the following: IRS market t market quoted par swap rate (*) 5.00% 5.97% 6.91% implied zero swap rate, r 1 (0,t) 5.00% 6.00% 7.00% (*) Assume all swaps have floating legs with annual payments. Then, check that the Z-spread for bond C is z = 2.13%. For bonds with embedded options, the spread needs to be adjusted for the value of the option. This requires using an option pricing model to allow for stochastic interest rates. Definition 5.1.4: Option-Adjusted Spread The Option-Adjusted Spread (OAS) on a bond with an embedded option is related to the Z-spread and option value as follows: 1. For a callable bond(or bonds with prepayment options, like most mortgage-backed securities), 2. For a putable bond, where Z-spread(%) = OAS(%) + Option Value (%) Z-spread(%) = OAS(%) - Option Value (%) the Z-spread is computed as in (5.1) the embedded Option Value, in %, is computed with an option pricing model. Intuitively, 1. A callable bond is bad for the investor 1, so the observed Z-spread must be high to compensate the investor: Z-spread > OAS. 1 If interest rates fall, the issuer will call the bond, forcing the investor to reinvest at lower rates.

68 5.1. Credit spreads A putable bond is good for the investor, so he will hold it at an observed low Z-spread: Z-spread < OAS. Since the Z-spread is computed with the stated full stream of nominal coupons, it does not properly account for the option. The OAS is a cleaner measure of the extra return that the investor expects to receive, given that there is some probability that the option will be exercised. The OAS is thus a better measure to compare different bonds/issuers when some of the bonds have embedded options. Figure 5.1 shows a Bloomberg screen with all these spreads for a bond issued by EDP. Figure 5.1: Bond spreads in Bloomberg Pricing The simplest approach to price a particular bond is just to add a credit spread to the appropriate reference rate. We typically use spreads from other comparable securities, so this is called relative valuation. Some possible alternatives to estimate the credit spread: 1. If there are already other bonds issued by the same firm, use anaverage spread from those bonds. 2. If the bond or the firm is rated, use the average credit spread for comparable bonds with the same rating. This amounts to pricing the bond using the yield curve for the comparable rating class.

69 5.2. Pricing with objective probabilities of default If there are Credit Default Swaps traded on the firm, use the CDS spread as a Z-spread. 4. If there is no market information, of if the analyst thinks the market is wrong, he can make his own estimation of the spread, perhaps based on a structural model of credit risk (like Merton s model). Check EXCEL BVAL EDP.xlsx: YTM, FMC While this approach has the advantage of providing a quick and easy approximation to the price, it is too simplistic for providing precise trading signals. It is not clear how to adjust the credit spread for all the factors that affect the recovery rate of the particular bond(subordination, collateral, etc), which is critical for pricing defaultable bonds. Furthermore, it is not clear how to incorporate a term structure of default probabilities. Therefore, the relation between price and spread is more commonly used in the other direction: given the market price, we can compute the implied credit spread. This implied credit spread is then useful as a measure of the premium that we get for taking the credit risk of a particular bond. Credit spreads are a more meaningful representation of credit market conditions than corporate bond prices or even yields. 5.2 Pricing with objective probabilities of default To account for the specific bond recovery rate and term structure of default probabilities, we can model the bond s cash flows with a tree. Suppose that the bond has 3 years to maturity and the coupons are paid annually. Further assume that default can only happen at the coupon payment dates and, in case of default, we recover a fraction R of notional plus accrued interest. The cash flows are represented in figure 5.2.

70 5.2. Pricing with objective probabilities of default 70 Figure 5.2: Cash-flow tree for a defaultable bond Using objective, or real-world, or physical, or measure P probabilities of default, the price is the following.

71 5.2. Pricing with objective probabilities of default 71 Proposition 5.2.1: Defaultable bond pricing with objective probabilities The price of a defaultable coupon bond, assuming that default can only happen at the coupon payment dates t 1,...,t m, is V 0 = m i=1 EP [Cash Flow at t i ] Z d (0,t i ), or V 0 = m C i Z d (0,t i ) PS P (0,t i ) i=1 + m (1+c)R Z d (0,t i ) PS P (0,t i 1 ) PD P (t i 1,t i ) (5.2) i=1 where Z d (0,t i ) is the risky discount factor appropriate for this defaultable bond (but see the remark below) PS P (t,t) is the objective (measure P) probability of survival between time t and T. PD P (t,t) is objective (measure P) probability of default between time t and T. c is the coupon rate and C i is the total payoff at time t i R is the recovery rate Possible alternatives to estimate the objective probabilities of default: 1. Rating transition matrices 2. Internal credit scoring models (eg, logit-type models) 3. Estimations based on structural models of credit risk, like Moody s KMV EDF. Remark. The difficulty with this approach is that we need to specify the risky discount factors Z d (.). These Z d (.) should be high enough to compensate for the investor s risk aversion, but it is not obvious exactly how much should they be. Notethatwe cannot just addamarket credit spread (like the

72 5.3. Risk-neutral probabilities of default 72 z-spread estimated with best case cash flows) to risk-free rates to get the Z d (.) because then (5.2) would be double-counting the possibility of default Risk-neutral probabilities of default Definition Risk-neutral, or market-implied, or measure Q probabilities of default are probabilities that are implied by the market prices of defaultable bonds. Consider a one-period zero-coupon bond. Under the risk-neutral pricing setup, its current price, V 0, must be the expected payoff discounted at the risk-free rate (r): V 0 = EQ 0[V 1 ] 1+r = (1 q)+qr (5.3) 1+r where q is the risk-neutral default probability and R is the recovery rate. This is called risk-neutral pricing because we don t see the usual risk correction in the denominator; instead risk aversion is impounded in the probability q. The risk-neutral q is higher than the physical (or historical) p because it is loaded with default risk premium (q > p) Estimation from corporate bonds Given a set of market prices of corporate bonds from a given issuer, we can do reverse engineering to back out the probabilities that must have been used to determine those prices. 1 Period Given the market price, V 0, of a one-period zero-coupon bond, we can solve (5.3) directly for q. However, it is more convenient to work with spreads. Note that the same market price, V 0, can be represented by the usual discounting where creditrisk aversion is fully incorporated in the denominator, using best-case cash flows: 1 V 0 = (5.4) 1+r +s 2 For an investment bank, a first approximation may be the bank s cost of capital, though this does not take into account the risk of the specific bond.

73 5.3. Risk-neutral probabilities of default 73 where s is the credit spread. Equating the two expressions for V 0 we get (1 q)+qr 1+r = 1 1+r+s which can be solved explicitly for the default rate q = s (1+r+s)(1 R) (5.5) Example What is the probability of default of the issuer of a zero-coupon 1-year bond trading with a spread of 80 b.p., if the expected recovery rate is 40% and the risk-free interest rate is 5%? q =... = 1.26% For intuition, note that: 1. When interest rates are not too large, we have (1+r+s) 1. Hence, s q = (1+r +s)(1 R) s 1 R 2. This implies that s q(1 R) = q LGD 3. This is intuitive: the credit spread is approximately equal to the (riskneutral) expected loss. Extension to T Periods Given a set of risky coupon bonds of an issuer, we can estimate the implied zero-coupon rates by one of the usual methods (bootstrap, Nelson-Siegel, etc). Let s(t) denote the spread between the risky and the risk-free rate for maturity T. The accumulated probability of default over T periods comes from the natural extension of the 1-period case: q(0,t) : 1 q(0,t)+q(0,t) R [1+r(0,T)] T = 1 [1+r(0,T)+s(T)] T (5.6)

74 5.3. Risk-neutral probabilities of default 74 Example Consider the following zero-coupon rates(ear, in %) and the resulting spread for rating A: Rating Category T=1 year T=2 years Risk-free Rating A Spread for rating A The Recovery rate is 40%. 1) Compute the market-implied default probability for the first year. q(0,1) =...= ) Compute the cumulative q(0,2) q(0,2) =... = 6.53% 3) Compute the market-implied default probability during year 2, ie, the conditional q(1,2). Recall that q(0,2) = 1 P[survival(0 to 2)] = 1 (1 q(0,1)) (1 q(1,2)) Hence, q(1,2) =... = % There are commercial providers of risk-neutral probabilities. For example, FreeCreditDerivatives.com publishes market-implied marginal default probabilities see figure 5.3. For each rating class, there is a term structure of default probabilities. Note further how there is one such term structure for each recovery rate Risk-neutral versus objective probabilities Applications Risk-neutral q probabilities are useful for pricing credit derivatives and bonds. For example, we can estimate RN probabilities from bonds of a given firm and then use them to price credit derivatives on the same firm (or vice-versa). Objective p probabilities are useful for managing credit risk (eg, determining capital reserves, computing Credit Value-at-Risk).

75 5.3. Risk-neutral probabilities of default 75 Figure 5.3: Market-implied default probabilities from FreeCreditDerivatives.com

76 5.3. Risk-neutral probabilities of default 76 Magnitudes Risk-neutral probabilities have the job of accounting for credit risk. Hence, the risk-neutral probability of the bad default outcome has to be higher than the true probability: q > p. It is interesting to compare the implied default probabilities with historical frequencies of default. The surprising result is that market implied rates are much higher than historical rates: q market implied >> p historical. Typically, implied rates tend to be higher by a factor of 2 3. This in turn implies that the credit spreads in the market seem too large (recall s market LGD q market implied ). In other words, the credit spread should compensate for defaults so that an investment in risky bonds would just break even relative to risk-free bonds. However, we find that corporate bonds are providing a compensation that seems more than adequate for their level of credit risk (for reasonable levels of risk aversion). This empirical fact is called the credit spread puzzle. For instance, Huang and Huang (2012) find that, for investment grade bonds, historical default loss data can only explain a small fraction around 20% of the observed corporate-treasury yield spread. Figure 5.4 illustrates the puzzle. The difference in spreads may be partially justified by other factors, such as liquidity. Additionally, it may be the case that investors are properly pricing rare but devastating events, such as the 2008 crisis. 3 3 See chapter 8 in Servigny and Renault (2004) for an overview of credit spreads.

77 5.4. Default intensity 77 Figure 5.4: Credit Spread Puzzle Source: Huang and Huang (2012, figure 2) 5.4 Default intensity The standard hypothesis in reduced-form models is that the number of defaults (N) during a time interval is a random event following a Poisson distribution. Let λ denote the default probability per unit of time (typically, one year). This quantity is called the default rate, the default intensity, or the default hazard rate. From the definition of the Poisson distribution, the probability that k events happen during t is P[N = k] = e λ t (λ t) k k! for k = 0,1,2,... Therefore, the survival probability is given by the probability that no default event occurs: PS(t,t+ t) = P[N = 0] = e λ t In general, we can have different default intensities for different periods. In the limit, we can have a different intensity for each instant of time. Let

78 5.5. Risk-neutral pricing 78 λ(t),t 0, be the default probability per unit of time (typically, one year) for time t. It can be shown that the survival probability from t to T is given by 4 { T } PS(t,T) = exp λ(s) ds t In particular, note that if the default rate is again constant, λ(t) = λ, t, we have PS(t,T) = e λ(t t). Example Suppose that the constant default intensity is 5% per year. Then, the one-year survival probability is PS(0,1) =... = 95.12% and the one-year default probability is PD(0,1) =... = 4.88% We can also easily get the probabilities for other terms. For instance, the cumulative 2-year survival probability is PS(0,2) =... = 90.48% The 1-month default probability is PD(0,1/12) =... = 0.42% 5.5 Risk-neutral pricing Consider again the representation of the bond s cash flows in a tree like figure 5.2. Using risk-neutral probabilities, we can easily price the bond by discounting all the expected cash flows at the risk-free rate. 4 Alternatively, the probability of defaulting within the interval [t,t+dt) conditional on surviving up to time t is PD(t,t+dt) = P[τ < t+dt τ t] = λ(t)dt, where τ represents the default time.

79 5.5. Risk-neutral pricing 79 Proposition 5.5.1: Defaultable bond risk-neutral pricing The price of a defaultable coupon bond, assuming that default can only happen at the coupon payment dates t 1,...,t m, is V 0 = m i=1 EQ [Cash Flow at t i ] Z(0,t i ), or V 0 = m C i Z(0,t i ) PS Q (0,t i ) i=1 m + (1+c)R Z(0,t i ) PS Q (0,t i 1 ) PD Q (t i 1,t i ) (5.7) i=1 where Z(0,t i ) = e r(0,t i)t i is the risk-free discount factor ( PS Q (t,t) = exp ) T λ(s)ds is the risk-neutral (measure Q) t probability of survival between time t and T. λ(t) is the risk-neutral default intensity for time t PD Q (t,t) = 1 PS Q (t,t) is the risk-neutral (measure Q) probability of default between time t and T. c is the coupon rate and C i is the total payoff at time t i R is the recovery rate Remarks: The risk-neutral default intensity, λ(t), and the corresponding probabilities (PS and PD), can be estimated from: 1. market prices of other bonds with similar risk (bootstrap procedure similar to the one in section 5.3). 2. CDS spreads Formula (5.7) uses accumulated survival probabilities while the tree in figure 5.2 uses marginal probabilities. Note that they are related by PS(0,T) = PS(0,t) PS(t,T), for 0 < t < T Note that the last term in (5.7) is the probability of defaulting in the period after having survived until the beginning of that period. It can

80 5.5. Risk-neutral pricing 80 also be computed as 5 PS(0,t i 1 ) PD(t i 1,t i ) = PS(0,t i 1 ) PS(0,t i ) Example Compute the price of a 2-year bond, paying annual coupons at 7%, under the following assumptions: The risk-neutral default intensity is estimated to be: { 0.01, 0 < t 1 λ(t) = 0.02, 1 < t 2 The recovery rate is estimated to be 40%. The risk-free curve is flat at r (0,T) = 3%, T SOLUTION: PS Q (0,1) =... = 0.99 PS Q (0,2) =... = PS Q (0,0) PD Q (0,1) = 1 (1 PS Q (0,1)) =... = 0.01 PS Q (0,1) PD Q (1,2) = PS Q (0,1) PS Q (0,2) =... = and using (5.7), V 0 =... = % General pricing formula. If the firm has other liabilities, bankruptcy can happen at dates other than the coupon-payment dates for this particular bond. The general pricing formula is (formula 9.18 in Chaplin (2010)) V 0 = m C i Z(0,t i ) PS Q (0,t i ) i=1 + T where A(s) is the accrued interest at time s. 0 (1+A(s))R Z(0,s) PS Q (0,s) λ(s) ds 5 { Note that PS(0,t i 1 ) PD(t i 1,t i ) = PS(0,t i 1 ) (1 PS(t i 1,t i )) = PS(0,t i 1 ) exp } { t i 1 0 λ(s) ds exp } { t i t i 1 λ(s)ds = PS(0,t i 1 ) exp } t i 0 λ(s)ds = PS(0,t i 1 ) PS(0,t i )

81 5.6. Asset Swaps Asset Swaps Definition Definition 5.6.1: Asset Swap An Asset Swap is a package that includes: One Corporate bond (with fixed coupon); One Interest Rate Swap, with the same maturity as the bond, to exchange the bond fixed coupon for a floating rate. Suppose bank ABC buys from bank XYZ an Asset Swap on a Corporate bond C. This corresponds to two simultaneous trades: 1. ABC buys from XYZ the corporate bond with fixed-rate coupon c. 2. ABC enters into a IRS with XYZ, where ABC pays the fixed leg c and receives variable Libor +s as. The net effect is to transform the fixed-rate defaultable C into a floating-rate defaultable C. See figure 5.5. Figure 5.5: Asset Swap Remarks:

82 5.6. Asset Swaps 82 The IRS is not affected by credit events on C. Even if C defaults, bank ABC still has to pay to XYZ the fixed coupon c. If default happens, the IRS can be unwind at its market value. 6 Banks use Asset Swaps to manage interest rate risk, i.e., to match their assets and liabilities (typically floating rate). Asset Swaps are very liquid instruments and it is sometimes easier to buy the AS package than the corporate bond by itself Asset Swap pricing The asset swap spread s as is determined so that the price of the asset swap package equals the par value of the corporate bond. 7 Proposition 5.6.1: Asset Swap Pricing The spread in a par Asset Swap, s as, is such that: where P := market price of the bond. A irs := 1 [1+r irs (0,1)] [1+r irs (0,T)] T V irs := P = V irs s as A irs (5.8) s as = V irs P A irs (5.9) c c [1+r irs (0,1)] [1+r irs (0,T)] T c := bond coupon r irs (0,t) := zero-coupon swap rate for maturity t. Proof. We want s as such that the fair price of the asset swap package equals the par value of the corporate bond. That is, we pay 100% for a bond that is trading at P. Therefore, the IRS value must compensate this gain (or loss) 6 There are also clean asset swaps where the IRS disappears if C defaults. 7 This is the standard par asset swap. There are also market asset swaps.

83 5.6. Asset Swaps 83 in the bond. More formally, Gain in Bond+Value IRS = 0 (P 100%)+Value IRS (receive floating Libor + s as, pay fixed c) = 0 (P 100%)+Value IRS (receive floating Libor, pay fixed c s as ) = 0 [ c s as (P 100%)+ 100% [1+r irs (0,1)] 100+c s ] as = 0 [1+r irs (0,T)] T P = V irs s as A irs Note that we do not use risky discount factors to price the IRS. This is because the IRS does not knock out on default of the bond. Example Bond C is a 3-year bond with 10% annual coupons trading at %. The IRS rates are: IRS market t zero-coupon rate, r IRS (0,t) 5% 6% 7% Compute the AS spread on this bond. V irs =... = % A irs =... = s as =... = 2.14% To understand the AS spread, suppose we buy the AS: 1) On the one hand, we are getting a good deal on the bond because we are paying 100% for a bond that is trading at P = %, i.e., we gain 2.53%. 2) On the other hand, in the IRS we are paying c s as = = 7.86%. If we went directly to the IRS market, we would pay 6.91% fixed for 3 years (x : 1 = x/ x/ (1 + x)/ x = 6.91%). We thus pay an excess of 0.95% in each of the next 3 years, which has a present value of 2.53%. This is exactly the same as we gained on the bond! The s as is usually positive, but it is possible for entities with high rating to have a negative spread.

84 Chapter 6 Credit Derivatives 6.1 Introduction Credit Derivatives (CD) are financial contracts that essentially provide insurance against deterioration in credit quality of a specified issuer. The simplest and most important contract (CDS) works exactly like an insurance policy, with regular premiums paid by the protection buyer to the protection seller, and a payout if a specified credit event occurs. The specified issuer, also designated by name, is typically not a party to the credit derivatives contract. Also, the maturity of the derivative is typically different from the maturity of the underlying asset Purpose Credit derivatives isolate credit risk from other sources of risk (interest rate risk, liquidity risk), and then allow the transfer of this credit risk. Hence, CDs serve one of two purposes: Reduce credit exposure. If bank a bank cannot sell assets for some reason (client relationship, taxes, liquidity), he can buy credit derivatives to eliminate credit exposure to some of those assets. The CD will payoff if the obligors default. Commercial banks typically enter as protection buyers in CD to eliminate some of the credit risk in their portfolios; 84

85 6.1. Introduction 85 Increase credit exposure. If an investor is unable or unwilling to buy assets directly, he can sell credit derivatives to obtain credit exposure. He provides credit protection for a fee, which enhances his return (if no major credit events happen). Insurance companies, Hedge Funds, Private investors, and other financial institutions are the natural protection sellers in CD Credit Event Credit derivatives pay-off when a credit event occurs. That is, a credit event is what triggers the payment by the protection seller to the protection buyer. What constitutes a credit event is defined specifically in the legal documents that describe the CD contract. Some credit events that may be specified in the contract are (ISDA definitions): Bankruptcy or Liquidation of the company. Failure to Pay: default on coupon or principal payment obligations. Repudiation or Moratorium: refusing to pay or delaying payment (typical in sovereign defaults, also called technical default). Restructuring: change in the terms of an existing loan. Includes reduction and deferral of interest or principal. Downgrade in S&P or Moody s credit rating below a specified minimum rating. Increase in credit spread above a specified maximum rate Market characteristics The market for CD started in the early 1990s and by 2000 the notional value of the market was over $800 billion. Figure 6.1 shows the most used credit derivatives in The market grew exponentially until Credit Default Swaps are by far the most important credit derivative: in 2007 the global notional

86 6.2. Credit Default Swaps 86 Figure 6.1: Notional value by derivatives type (Risk survey, 2000) outstanding of CDS alone exceeded 40 trillion dollars. Since the crisis, the market has cooled off a bit and the notional outstanding is around 24 trillion dollars in Credit derivatives areover-the-counter (OTC) products. 2 Therefore, they can be tailored to meet specific needs of clients, which has contributed to the growth in CD use. However, since the financial crisis there has been some regulatory pressure to force CD to trade only on exchanges. 6.2 Credit Default Swaps Definition The most common credit derivative is the Credit Default Swap (CDS), sometimes abbreviated to credit swap or default swap. The structure of a typical CDS is illustrated in figure Since CD are traded over the counter, data is scarce and comes mostly from surveys by the BIS, ISDA, Risk, and BBA. 2 There are some centralized trading platforms like Creditex and CreditTrade, but they serve only to bring the counterparties together; in the end, the contract is signed between the two counterparties.

87 6.2. Credit Default Swaps 87 Figure 6.2: Credit Default Swap Payments. The party buying credit protection pays a fixed fee periodically (typically quarterly or semi-annually) or/and an upfront premium (typically for distressed underlying assets), usually negotiated as a number of basis points times a notional amount. The first fee is usually payable at the end of the first period. These coupons are paid until the specified maturity of the CDS or until a credit event happens. The party selling protection makes no payment unless the credit event happens. Credit event. The typical credit events that may be specified in a CDS are: bankruptcy of the reference entity (underlying issuer), failure to pay (default) on the underlying asset (bond or loan), or restructuring. Underlying asset. The type of asset used to determine whether a credit event has occurred. One common definition is Borrowed Money, which includes any obligation of the reference entity to return borrowed money. Instead, a specific bond may be defined in this case named Reference Obligation. Settlement. If default does indeed happen, there are two possible types of settlement: Cash settlement. The default payment is the notional amount minus a recovery value. (For bonds, the recovery value is their market value after default, though the bond may be very illiquid and the price volatility may be high following default.)

88 6.2. Credit Default Swaps 88 Physical settlement. The buyer delivers the underlying asset to the seller, in return for the face value of the underlying. This method eliminates the uncertainty of the valuation process. The protection buyer can deliver any obligation not subordinated to the reference obligation. If nothing else is specified in the contract, Senior Unsecured is assumed. In either case, the protection buyer receives from the swap the amount he lost on the underlying. Note how a CDS works exactly like insurance Usage The most common CDS specification has a 5-year maturity, a notional amount between $5 $20 million, and is written on a large firm at the lower end of the investment grade rating scale (A to BBB). The standard use of a CDS, from the buyer s perspective, is to eliminate credit risk. Example risk. 4 This example illustrates how a CDS hedges credit Initial trade. Party A enters as protection buyer and B as protection seller in the following CDS: - Reference Entity: Daimler Chrysler - Reference Obligation: Bond XYZ issued by Daimler Chrysler - Credit event: Failure to pay on the reference obligation - Term of the CDS: 5 years - Notional of the CDS: 20 Million USD - CDS fee: 116 bp, semi-annual payment - Physical settlement. Fee payments. At the end of each semester, A pays to B M$ = $116, (the standard day-count method is Act/360). These payments stop as soon as a credit event occurs. 3 Some people say CDS were not instead called Default Insurance just for regulatory reasons. 4 This is from example 2.3 in Schönbucher (2003)

89 6.2. Credit Default Swaps 89 Default payment. Suppose after 8 months Daimler Chrysler fails to pay a coupon that was due on the reference bond. First, A pays the accrued fee : $116, = $38,667 Second, A delivers Daimler Chrysler bonds to B with a total notional value of 20 M$ (the notional value of the CDS). Third, B pays 20 M$ to A. In addition, CDSs can be used to: Reduce concentration risk. Banks can manage the concentration risk within their credit portfolios by buying CDS and protecting themselves against default by some of their clients. Create credit investments. By selling CDS, financial institutions are in fact creating non-funded and off-balance-sheet credit investments. Hedge trade receivables. Corporations may use credit derivatives to reduce overexposure to customer s credit risk. Free up credit lines. CDS are a very efficient way to free up credit lines, as in example Example Suppose bank A has lended $100m to corporation X. Bank A needs to keep aside 8% of the $100m as economic credit capital to absorb credit losses. Suppose dealer B offers the following CDS: annual fee = 20 bp * $100m; $8m payment to bank A if corporation X defaults on the loan. By buying this CDS, bank A is able to free up $100m for new business with X at no additional capital charge. The cost of freeing up this capital is 20 basis points, i.e., $0.2m per year. 6 Figure 6.3 illustrates this operation. 5 Guarantees issued by or protection provided by entities with a lower risk weight than the counterparty exposure is assigned the risk weight of the guarantor or protection provider. (Basel II, page 49, Article 141) 6 Of course, the dealer is trying to sweeten the terms of the CDS. The real cost to the bank is 0.2m/8m = 250 basis points.

90 6.2. Credit Default Swaps 90 Figure 6.3: Using a CDS to free up capital Indexes In addition to single-name CDS, there are also CDS indexes. The indexes measure the (equal-weighted) average premium (ie, credit spread) in a group of CDS. Prices are quoted daily by leading investment banks. The indexes are used by asset managers, hedge funds, insurance companies, etc, to hedge or create credit exposure. The indexes trade like single name CDS and are thus highly liquid. Two important indexes are: itraxx family. Launched in 2004 by International Index Company (intindexco.com), now run by Markit. Includes several indexes for the European market and for some Asian countries. The benchmark index is the itraxx Europe, composed by the top 125 European names in terms of CDS volume. A new series of the index (eventually based on a different underlying portfolio) is issued every 6 months (March and September). For example, itraxx Europe Series 5 was issued in Mar/2006. Some of the names in that series were Banco Comercial Portugues, Banco Espirito Santo, and Energias de Portugal. There are indexes for 3, 5, 7, and 10-year maturities.

91 6.2. Credit Default Swaps 91 Other important itraxx indices are the itraxx Europe HiVol (30 widest spread names), the itraxx Europe Crossover (50 most liquid subinvestment grade entities), the itraxx Asia ex-japan IG (40 investment grade names). CDX family. Launched in 2003 by Dow Jones Indexes and now run by Markit (markit.com). The CDX family includes several indexes for segments of the North American market and for Emerging Markets. For example, CDX.NA.IG is composed by top 125 North America investment grade names. Also reconstituted every 6 months (Mar and Sep). CDX4 included Ford and GM, but CDX5 dropped them because they were no longer investment grade. Other important indices are the CDX.NA.IG.HVOL (30 names with highest volatility from CDX.NA.IG), CDX.NA.HY (High Yield, 100 names), CDX.EM (14 names from Emerging Markets like Latin America, Eastern Europe, and Asia), CDX.EM.DIVERSIFIED (40 names from Emerging Markets). Figure 6.4 shows the evolution of the itraxx Europe and Crossover. At each point in time, we use the value for the most recent series of the index. As discussed in section 6.3, the spreads can be interpreted as the market premium for taking credit risk. For example, the itraxx Crossover spread plus the Government bond yield should be reasonably close to the average yield on a sub-investment grade bond. The indexes are thus a good measure of credit market conditions. Figure 6.4: itraxx Europe and Crossover

92 6.3. CDS pricing 92 Example Hedge Fund A believes the macroeconomic outlook for the Euro economy is good and thus wants to gain exposure to credit risk in that market. Using itraxx Europe, the fund sells protection on a notional of 10Me. The itraxx spread is 0.40%. If no credit event happens, the market maker pays to fund A 40 bps per annum quarterly on a notional amount of 10Me. Suppose one of the reference entities defaults in year 3. Recall that each name weights 1/125=0.8% in the index. Hence, Fund A pays to market maker 0.8% 10M = EUR; the market maker delivers to A EUR face value of Deliverable Obligations of the reference entity (physical settlement). After this, the market maker continues to pay 40 bps, but on a smaller notional: ( ) 10M = EUR. 6.3 CDS pricing We start by using arbitrage arguments to relate the CDS spread to the credit spread on the underlying corporate bond and to the asset swap spread. Even though these relations only give an approximate CDS price, they are free from assumptions and therefore are robust approximations to CDS prices. We then present a full model to price CDS. This requires more assumptions, but provides a more precise price for both new and existing CDS Triangular Arbitrage Consider two portfolios: 1. Portfolio I: One risk-free (default-free) bond B, with coupon r f 2. Portfolio II: One risky, defaultable, corporate bond C, with coupon c. One CDS on this bond, with spread s cds

93 6.3. CDS pricing 93 All instruments have the same notional and maturity, and both bonds are priced at par (hence the ytm equals the coupon). Portfolio II can be seen as a synthetic default-free security since the CDS hedges the default risk. Therefore, the two portfolios must offer identical returns; otherwise there are arbitrage opportunities (buy the cheaper and sell the more expensive portfolio). Hence: long B = long C + long CDS r f = c s cds s cds = c r f Example We can invest $1m in a Treasury Bond offering a return of r f = 5%. 2. Alternatively, we can invest $1m in a corporate bond offering a return of c = 8%. 3. To avoid arbitrage, the CDS spread on this corporate bond must be s cds =... However, for this arbitrage relation to hold exactly, the payoffs of the two portfolios would have to be exactly the same in both survival and default states. They differ in default: Portfolio I pays the market price of the riskfree bond, while Portfolio II pays the notional value of the corporate bond: Cash flows for two portfolios State Portfolio I Portfolio II I-II B C CDS Initial t = Survival: t = 1,...,T r f c s cds r f (c s cds ) t = T Default t = τ B(τ) Recovery 100-Recovery B(τ) 100 The value of the default-free bond may be different from par at default (B(τ) 100) because interest rates move stochastically. Hence, we only have an approximate no-arbitrage relation: s cds c r f

94 6.3. CDS pricing 94 The CDS spread thus equals the credit spread of the bond (we are assuming that the bond is priced at par, so c = YTM). The risk-free rate can be proxied by the Treasury rate (G-spread) or, as a better alternative, by the LIBOR/IRS curve (I-spread). 7 Typical deviations from this relation are of the following order: 8 (c r Treasury ) s cds 63 bps (c r IRS ) s cds 7 bps Even though triangular arbitrage pricing is often imprecise, it is very important in practice because it provides an approximate no-arbitrage price that is independent of any pricing model Pricing CDS with Asset Swaps Motivation: triangular arbitrage with floaters Suppose we wanted to price a CDS on a par floating-rate corporate bond. Consider two portfolios: 1. Portfolio I: Onerisk-free(default-free)floatingratebond B, withcouponr f = Libor. This bond trades at par, B(0) = 100. (We can also think of this as a bank deposit at rate Libor). 2. Portfolio II: Onerisky, defaultable, corporatebond C,withcouponc = Libor+ s corp. The spread is such that the bond trades at par C(0) = 100. One CDS on this bond, with spread s cds All instruments have the same notional and maturity and both bonds are priced at par. 7 The LIBOR/IRS curve is a better proxy for pure default-free rates than the Treasury curve. Treasury rates are lower than LIBOR/IRS rates mostly because of a convenience yield from holding Treasuries, not because of credit risk. For details, see Hull (2012, sec 4.1) or my teaching notes on Fixed Income. 8 These numbers are from Alan White s presentation at Moody s KMV 2006 Credit Risk Conference.

95 6.3. CDS pricing 95 Recall that a floating rate bond trades at par at the coupon payment dates. Hence, in this case we removed the interest rate risk existing in triangular arbitrage with fixed rate bonds (previous section). The two portfolios thus have very similar cash flows: Cash flows for two portfolios State Portfolio I Portfolio II I-II B C CDS Initial t = Survival: t = 1,...,T Libor Libor +s corp s cds s cds s corp t = T Default t = τ B(τ) 100 Recovery 100-Recovery 0 Note that there still may be a small mismatch if default happens between coupon payment dates: B(τ) will be slightly above 100 due to accrued interest, while Portfolio II will only pay 100 because the CDS does not protect accrued interest. Still, the error is small. To avoid arbitrage, we thus have that the CDS spread must be very close to the coupon spread on the corporate floater: s cds s corp Typically we want to price CDS on fixed-rate bonds, not floaters. Therefore, we use a proxy: Asset Swaps. The relation between AS and CDS prices We can use the Asset Swap instead of a defaultable floater and construct the same arbitrage strategy as before. Consider two portfolios: 1. Portfolio I: Onerisk-free(default-free)floating-ratebond B,withcouponr f = Libor. This bond trades at par, B(0) = Portfolio II: One Asset Swap package on a risky, defaultable, fixed-rate corporate bond C. The net cash flow of the asset swap is Libor +s as.

96 6.3. CDS pricing 96 One CDS on bond C, with spread s cds All instruments have the same notional and maturity and both bonds are priced at par. Cash flows for two portfolios State Portfolio I Portfolio II I-II B AS CDS Initial t = Survival: t = 1,...,T Libor Libor +s as s cds s cds s as t = T Default t = τ B(τ) 100 Recovery + value of IRS(τ) 100-Recovery }{{} 0 0 Note that there still may be a small mismatch if default happens between coupon payment dates: B(τ) will be slightly above 100 due to accrued interest. Furthermore, the IRS component of the Asset Swap has to be unwind at market value. However, the expected value of the IRS at default is close to zero if the initial value of the underlying bond is at par and interest rate movements and defaults occur independently (see Schönbucher, 2003, p.32, for details). Hence, these errors are typically very small. To avoid arbitrage, we thus have that the CDS spread must be very close to the Asset Swap spread: s cds s as Basis. The difference between the CDS spread and the Asset Swap (using bid quotes) is called Basis: Basis = s cds s as The basis is typically in the order of 10 bps. 9 A much larger basis means that the default risk of the underlying obligor is priced differently in the CDS market and in the bond market. It signals a possible arbitrage opportunity between the CDS and the Asset Swap. 9 In the case of distressed debt ( fallen angels ) the basis is much larger. See Schönbucher (2003).

97 6.3. CDS pricing 97 Figure 6.5: Portugal Telecom (Source: Reuters) CDS spread vs. Z-spread Recall that the Z-spread (z) is determined through the following equation (see eqn 5.1): P = c [1+r irs (0,1)+z] c [1+r irs (0,T)+z] T where P := market price of the bond. r irs (0,t) := zero-coupon rate for maturity t from the swap curve.

98 6.3. CDS pricing 98 Example Bond C is a 3-year bond with 10% annual coupons trading at %. The IRS rates are: IRS market t zero-coupon rate, r IRS (0,t) 5% 6% 7% Check that z = 2.13% Traders use the Z-spread as a better approximation to the CDS spread: CDS vs. Bond The CDS spreadmust beclose to thez-spreadonthe underlying bond: s cds z Intuition. Compare with the asset swap formula (5.8), In the previous example this is P = V irs s as A irs % = 10% s as [1.05] + 10% s as [1.06] % s as [1.07] 3 s as = 2.14% While s as adjusts the numerator, the Z-spread adjusts the denominator. In practice, the Z-spread is usually a better approximation to the CDS spread because the Asset Swap is too sensitive to the coupon rate Pricing with a reduced-form model We follow O Kane and Turnbull (2003) and present the market standard model for pricing CDS positions to market. This is a reduced-form model because default is modeled as an exogenous random event. The model uses risk-neutral default probabilities estimated from market prices (of other CDS or bonds). Recall that the survival probability from t to T is given by { T } PS(t,T) = exp λ(s) ds t

99 6.3. CDS pricing 99 To simplify the notation in this section, we denote simply by P S(T) := PS Q (0,T) the risk-neutral probability of surviving from today (t = 0) until some future time T. The CDS matures in N periods. There are t 1,...,t N fee-payment dates. Fee leg The fee or premium leg is the series of payments made by the protection buyer. The present value of the fee payments is PV fee leg = s N DF(t i ) PS(t i ) t i i=1 where s is the CDS spread, DF(t i ) = e rt i is the discount factor for a risk-free rate r, t i = t i t i 1 is the number of years between payment dates (ie, the day count fraction between t i 1 and t i using the basis specified in the contract, eg, act/360). The previous expression ignores the effect of premium accrued from the previous payment date until the time of default. 10 If the accrual fee is paid upon default, O Kane and Turnbull (2003) show that the value of the fee leg is approximated by: N N PV fee leg = s DF(t i ) PS(t i ) t i +s DF(t i ) (PS(t i 1 ) PS(t )) t i i 2 where i=1 This is sometimes written as RPV01 := N i=1 i=1 PV fee leg = s RPV01 [ ] DF(t i ) t i PS(t i )+(PS(t i 1 ) PS(t i )) 1pa 2 (6.1) denotes the Risky Present Value of 1 bp paid on the premium leg, and the indicator 1 pa equals 1 if the contract specifies premium accrued (pa) and 0 otherwise. 10 This case is sometimes called an European CDS. Intuitively, it is as if default only happened at premium payment dates.

100 6.3. CDS pricing 100 Default leg The default, or contingent, or protection leg is the payment of (100% R) if a credit event happens (R is the Recovery Rate). The present value of the default leg is 11 PV default leg = (1 R) = (1 R) N DF(t i ) PS(t i 1 ) PD(t i 1,t i ) i=1 N DF(t i ) [PS(t i 1 ) PS(t i )] i=1 CDS spread We want to find the CDS spread that sets today s value of the CDS to zero (par CDS). Hence, the par CDS spread, s, is such that the two legs have the same value: PV fee leg = PV default leg or, 11 This formula is also an approximationwhen the loss payment may happen between fee dates(american CDS). However, O Kane and Turnbull(2003) show that the discretization error is typically small.

101 6.3. CDS pricing 101 Proposition 6.3.1: CDS pricing The CDS spread s is the value that sets the value of contract to zero: where s N i=1 [ ] DF(t i ) t i PS(t i )+(PS(t i 1 ) PS(t i )) 1pa 2 = (1 R) N DF(t i ) [PS(t i 1 ) PS(t i )] i=1 t 1,...,t N are the fee-payment dates DF(t i ) is the risk-free spot discount factor for maturity t i { PS(t i ) = PS Q (0,t i ) = exp } t i λ(s)ds is the probability of 0 survival for a risk-neutral default intensity function λ(t). t i = t i t i 1 is the number of years between payment dates (using the basis specified in the contract, eg, act/360) the indicator 1 pa equals 1 if the contract specifies premium accrued and 0 otherwise Note that if premium accrued is not paid, the equation is simply s N DF(t i ) PS(t i ) t i = (1 R) i=1 N DF(t i ) [PS(t i 1 ) PS(t i )] i=1 Example Consider a 1-year CDS with quarterly payments (t i = 0.25,0.50,0.75,1.00). Assume R = 0.4 and a constant λ = Premium accrued is due if default occurs between payment dates. The risk-free interest rate is flat at r = 5%. Check that the CDS spread is s = 60 basis points. (This can be easily done in a spreadsheet). Check EXCEL CDS pricing examples.xlsx

102 6.3. CDS pricing Market-implied default rates Given observed market CDS prices for different maturities, we can calculate the market-implied term structure of default rates. For example, we can use the 1-year CDS to estimate λ for the first year. Then, we can use this lambda and 2-year CDS to estimate λ for the second year, and so on. This process is known as bootstrap. The implied assumption is a piecewise flat term structure of hazard rates. Example Continuing the previous example, suppose we have one more CDS for the same firm. The market is thus: Maturity CDS spread (s) 1 yr 60 bp 2 yr 89 bp First, use the solver in Excel to find λ for the first CDS (from the previous exercise, we already know the result: λ = 0.01). Second, use again the solver to find λ for the second year. We want the model spread to match the market price of the2-yr CDS given that λ = 0.01 in the first year. That is, let the solver choose λ for the second year. The final result is λ(t) = { 0.01, 0 < t , 1 < t 2 Application. Once we have a term structure of default intensities for a given firm, we can very easily price any new CDS or bond issued by that firm. Check EXCEL CDS EDP.xlsx Mark-to-Market of an existing CDS We can now compute the mark-to-market (MTM) value of an existing CDS position. This is useful to measure the current profit/loss or if we need to unwind the CDS with the initial counterparty (the cash unwind value should equal the MTM value).

103 6.3. CDS pricing 103 Proposition 6.3.2: Mark-to-Market of existing CDS The MTM value of an existing CDS contract is where MTM (in $) = (s t s 0 ) RPV01 t V (6.2) s 0 := old spread fixed in the contract s t := current market spread for an equivalent CDS V := Notional dollar value of the contract RPV01 t is as defined in equation (6.1), with all variables computed with today s information. The MTM represents a potential loss or gain as follows: { > 0 gain for protection buyer, loss for protection seller MTM < 0 loss for protection buyer, gain for protection seller Example Some time ago we bought protection on the same name as the previous example for $10M notional at 70 bp and the CDS still has 2 years until maturity. Today, a 2-yr CDS on the same company is trading at 89 bp (as in the previous example). Intuitively, our MTM should be positive since we are paying too little for protection. Today, we have that for a 2-year fee leg (I computed this in an Excel spreadsheet): Our MTM value is thus MTM =... = $35,500 RPV01 = As expected, a long protection position has a positive MTM value.

104 6.3. CDS pricing Standardization and upfront payments The market is moving to standardized CDS, which may eventually be traded on exchanges. Some features that are being standardized include spreads and payment dates. The market quotes of standardized CDS are represented as prices instead of spreads (similarly to bonds). Standard spreads Some CDS are traded with standard coupon premiums, e.g., 500 bps regardless of the default risk of the underlying. In this case, the default risk of the underlying is accounted for through an upfront payment. Upfront payments have been common for high yield CDS for some time. The fair upfront payment corresponds to the Mark-to-Market value of the CDS (equation 6.2), redifining: s 0 := standard spread defined in the contract; s t := appropriate spread for the risk of the name. That is, Upfront Amount = (s t s 0 ) RPV01 t V If this value is positive, the buyer pays this amount to the seller at the beginning of the deal. Afterwards, he only pays premiums at s 0. This compensates the seller for receiving in the future spreads (s 0 ) that are too low (s t > s 0 ) for the credit risk of the name. Standard premiums and payments dates In addition to the coupon (or premium) values, the dates at which those coupons are paid are also being standardized. For example, some indices like the CDX.NA.HY already trade with standardized quarterly coupons on the 20th of March, June, September, and December. The conversion between a fair spread and a fair price depends on the pricing model and assumptions that we use. The next example follows the Markit CDS Converter, which has been developed in collaboration with ISDA The model is available at and the documentation at

105 6.3. CDS pricing 105 Example Today is 30/March/2011 and we are willing to pay a spread of 120 bps (s t ) to buy a given CDS. To convert this spread to a price, we use the Markit CDS Converter with the following inputs: 1) INPUTS CDS characteristics Description Value Remarks Maturity 20/Mar/ quarterly coupons remaining Trade date 30/Mar/2011 There are thus 10 days accrued Recovery 0.4 Running Coupon 100 bps Fixed coupon (s 0 ) Notional 10 MM (V) Risk-free Interest rates: Markit automatically inputs a risk-free curve. Use the rates in Figure 6.6. To simplify, assume these rates are continuously compounded rates and linearly interpolate as needed. Further, note that Markit only shows interest rates for the day when you access their page (1/Apr), not for the date we defined as trade date. Hence, there will be some slight differences between our results and theirs. 2) CALCULATIONS a) Using the model of the previous section, we find the constant default rate (λ) implied in the spread of 120 bps. Markit assumes that we are at the beginning of the coupon payment period, 20/Mar/2011. The result is λ = This implies a RPV01 = (I did this in Excel). b) The upfront ( MTM ), if we were at the beginning of the coupon payment period, would be Upfront Amount = $38, Since the upfront is positive, the buyer needs to pay this amount. c) Since we are 10 days past the last coupon, we compute the Accrued amount = days accrued 360 s 0 V = $2, The seller needs to pay this amount to compensate the buyer for the days without protection.

106 6.3. CDS pricing 106 d) The net amount that the buyer actually pays is thus the difference between the previous two values: Cash Settlement Amount = Upfront - Accrued Amt = $36, which is very close to the Markit value in red (figure 6.6). (Again, we are using slightly different interest rates). 3) PRICE QUOTE The price that would be quoted for this CDS would be a clean price, that is, not considering accrued premium (note that typically bond price quotes are also clean). It is computed in two steps: a) Define the upfront as a percentage of notional, Clean points upfront = Upfront Amt / V = % b) Compute the clean price as Clean Price = 1 - clean points upfront = % Which is extremely close to the Markit price (figure 6.6).

107 6.3. CDS pricing 107 Figure 6.6: Markit CDS Converter