Credit Risk Management: Pricing, Measurement, and Modeling

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Given the aims of this chapter, we will concentrate more on bank models explaining the different types of internal systems, the related parameters and the regulation issues. Chapter 5 discusses the definition, relevance, and application of loss given default LGD to credit risk management, as well as possible estimation approaches. LGD is one of the main parameters, along with probability of default PD and exposure at default EAD , of estimations for both the Basel II regulation and economic capital reporting purposes.

In such a context, severity models applied to open-default cases can suggest appropriate collection and recovery strategies that can lead to corrective actions on current criteria and the fine tuning of costs and resources. The internal validation of rating systems is part of the general framework for rating system controls. According to both Italian and international regulatory requirements, validation activities should not be confined to empirical validation methods and tests, but should assess the overall functioning of the rating system along different dimensions that include method, the IT system and data quality, and processes and governance.

Following the Basel Committee on Banking Supervision guidelines , p. A bank must demonstrate to its supervisor that the internal validation process enables it to assess the performance of internal rating and risk estimation systems consistently and meaningfully. The measurement of performance is of paramount importance to credit risk management, especially in retail credit risk management where the sheer number of decisions needs to be thoroughly controlled via a standardized approach and a consistent framework.

The high number of counterparties and decisions to be made calls for definite and possibly automated decision criteria and for ex post evaluation procedures concerning the rationality of the allocation of the limited capital available for the best investment alternatives on the basis of their expected perceived risk and return. Portfolio credit risk analysis is a relatively new field of study.

In this chapter, we compare different portfolio credit risk models that emphasize a common framework and we highlight how these models can be used for both regulatory and managerial purposes. In order to use the IRB approach for computing the credit capital requirement, Basel II requires banks to carry out a stress test analysis. In addition, stress testing clearly becomes very useful from a managerial point of view because it helps identify risk sources and define strategies to handle negative events.

Furthermore, the automation of the process of adjudication is possible for small credits and credit cards, by financial institutions through the application of credit scoring. The data items taken from bureau reports and applications are weighted and treated by credit risk scorecards. The items correspond to credit card application questions or credit bureau report entries and are referred to as characteristics by the credit industry. In addition to assessing whether the attribute is positive or negative, credit scoring will also assess by how much it is positive or negative.

There are three types of models listed and explained below consumer credit applications to be scored:. Each individual with accrediting history has their information contained in the credit bureau data. The types of data contained in each credit files are:. Individuals are allowed to obtain their own scores by some credit bureaus and be given some explanation of how their current scores can be improved.

Based on a series of some key variables, a more all-encompassing credit score can be derived by the application of a bureau score. Credit scoring models during their earlier development days were designed to put applicants in rank order based on their relative risk, due to the fact that lenders used the models to choose an appropriate cutoff score. The retail portfolios of banks should be segmented into sub-portfolios with similar loss features as required by the Basel Capital Accord. For these portfolios, both PD and LGD will have to be established by banks by segmenting each retail portfolio by score-bands each corresponding to a risk level.

Predicting applications that will turn out to be good or bad future risks is the purpose of credit scoring. By assigning high scores to good credits and poor scores to bad credits, distinguishing between the two is possible for the scorecard. The accuracy ratio AR , cumulative accuracy profile CAP , and its summary statistic are the traditionally applied validation methods. Finally, if a financial institution changes the nature of products offered to clients, then the scoring model should be replaced.

There is no restriction to the approximation of default in this risk modeling type. New areas have seen the application of credit scoring techniques with a number of different scores describing each customer. The sophistication in the interplay of risk and reward is being experimented, by leading banks, on how they can be accounted for. Risk-based pricing is particularly becoming popular in the market for credit products.

Market initiatives involve new clients being targeted and existing ones introduced to new products. Deciding the applications to be considered or rejected based on the scorecards is called screening applicants. By managing the account, decisions based on past behaviors or activities that were observed, can be made. These decisions range from: modification of credit line, product pricing, authorization of a temporary excess in the use of credit line, credit card renewal, and collection of past due interest.

The loop of customer relation cycle is summed up by cross-selling initiatives. As we shall see, it is also tractable and appealing as a foundation for modeling correlation in default times among various entities. Any CIR process is nonnegative. Because of the appearance of the square root in 3. Examples of Intensity Models 67 times called a square-root diffusion, and was introduced by Feller for applications in genetics.

The conditional survival probability p t , s implied by 3. Increasing volatility tends to cause a decline in the probability of default given survival to a given future date, other things being equal. Forward default rate in the CIR intensity model with varying intensity 68 3. Default Arrival form 3. This in turn increases the survival probability, meaning lower forward default rates, as illustrated in Figure 3. Mean reversion has the effect of reducing the impact of volatility on the shape of the curve of forward default rates.

Forward default rate in the CIR intensity model with varying meanreversion rate. Examples of Intensity Models Figure 3. The increments of Z over equal nonoverlapping equally sized time periods are independent and identically distributed, just as are those of a Brownian motion. As we saw in Section 3. Examples of Intensity Models 71 reversion at similar parameters. Similarly, increasing the variance of changes in intensity for these two distinct types of intensity models has similar effects. Further similarities are revealed in Chapter 5, where we consider the implications of these and other model types for term structures of credit spreads.

For this case, survival probabilities, similar to the univariate CIR form 3. One can allow correlation among the various elements of the state process X , as well as jumps, stochastic volatility, and many other features, while still maintaining the general solution form 3.

The theory is summarized in Appendix A. HJM Forward Default Rate Models It may be convenient in certain applications to suppose that default intensities are determined by a forward default rate model, in the spirit of the term-structure modeling approach of Heath, Jarrow, and Morton HJM Heath et al.

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For this, we assume that 72 3. One can likewise model forward default bond credit spreads with an HJM model, as reviewed in Appendix C. Default-Time Simulation Figure 3. Simulating a default time by the inverse-CDF method. Simulating a default time by the compensator method. Simulating the compensator. Statistical Prediction of Bankruptcy We have already reviewed several time-series models for describing aggregate frequencies of default, say among speculative-grade issuers.

In this sec- 3. Broadly speaking, we can classify models according to how the conditional probability of defaulting in the next period is modeled. We focus on three types of parameterizations: duration models, qualitative-response models, and discriminant analysis. A duration model typically parameterizes the forward default rate f t , normally called the hazard rate in the duration literature. In these simple duration models, the hazard rate f t depends only on t. More advanced duration models allow dependence of the hazard rate f t on a covariate vector Xi for each name i. Default Arrival For the proportional-hazards model 3.

Two widely studied special cases are the probit and logit models. Firms are dropped from the data after they default. So, as implemented here, discriminant analysis has all of the limitations of qualitative-response models e. For more on biases introduced by endogenous sampling and censoring effects, see Amemiya At the core of the differences among the three types of models are their implied default-time densities conditional on the sample Xi1 ,.

A duration model also allows for the calculation of the one-period 78 3. Default Arrival conditional surivival probability. For example, with the proportional-hazard model 3. More generally, if the covariates Xit form a stochastic process, then a Markov assumption for the covariates leads to estimates of conditional one-period survival probabilities. Example parameterizations were mentioned in Section 3. Comparing Prediction Methods In spite of the potential limitations of discriminant analysis outlined above, in practice, early studies found that it was approximately as useful as probit models in predicting bankruptcy.

Lennox also computed the rates of type-I errors a company fails but is predicted to survive and type-II errors a company survives but is predicted to fail , for various threshold points, of probit and discriminant-analysis models. The type-I and type-II error rates for his U.

Statistical Prediction of Bankruptcy Table 3. Interestingly, within the sample, Lennox found that the probit and discriminant-analysis models have comparable forecast errors. Thus, his results illustrate the importance of evaluating models for predicting bankruptcy on an out-of-sample basis.

Shumway criticized the static nature of probit and logit models, that is, their failure to account for the duration of survival. Table 3. Adding market variables market capitalization, excess equity return, and equity-return volatility to the information set further improves the forecasting performance see column Shumway-M. The last column of Table 3. This is automatically the case for duration models.

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For example, with the proportional-hazards model 3. Default Arrival Table 3. Shumway-M is the model that includes market variables as well. CJ-Shumway shows the results obtained by Chava and Jarrow using the Shumway model with a larger data set. For the qualitative-response and discriminant-analysis models, aging effects could be captured by introducing survival time as one of the conditioning variables in Xit.

There is no accounting for age in any form in these calculations. In an attempt to isolate any aging effects, Altman introduced the concept of a mortality rate, computed as the ratio of the total value of defaulting debt in a given year to the total value of the population of bonds at the start of this year. This was repeated for years 2, 3, and so on, from issuance for A-rated bonds. The entire process was repeated for each of the other credit ratings. Mortality rates were found to increase as the time since issuance increased for low-rated bonds but not for investment-grade bonds.

Similar aging effects were reported by Paul et al.

Recent evidence of this aging effect is shown in Figure 3. Moreover, the increase in mortality rates is particularly large for 3. Statistical Prediction of Bankruptcy 81 Figure 3. Realized default rates by original rating. Source: Altman and Kishore, CCC-rated bonds. After 3 years of seasoning, mortality rates tend to decline, though there is a spike in year 9 for CCC-rated bonds. A reason sometimes given for an aging effect on default rates is that many corporate bonds are callable. As time passes from the issue date, it becomes increasingly likely that the call option owned by the issuer will go into the money.

This may lead to a rising default rate as the time since issuance increases, merely from the selection bias caused by calling. This explanation would seem to be weakened, however, by the standard call-protection clauses in many corporate bond covenants. Whatever their sources, the aging effects displayed in Figure 3. They represent an average aging effect over the sample period, without taking account of either the business cycles or issuance cycles which may be correlated.

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Blume et al. In a sample of 82 3. Default Arrival bonds from the issue cohort of —, Blume and Keim show that the aging effect found in the original studies by Altman and Paul et al. Blume and Keim array default rates by year of default and age of bond. Cohorts of bonds are formed on the basis of the credit rating and year of purchase pair. The default experiences of these cohorts are tracked over time. The cohort consisting of B-rated bonds purchased in , for example, represents the default experience in the years following of all B-rated bonds that could be purchased in This would include not only bonds that were newly issued in but also seasoned issues that may or may not have had an original rating at issue of B.

Thus, the emphasis is not on time since issuance, as in the Altman studies, but rather on differences between cohorts indexed by calendar year and credit rating. Cyclical patterns in default, if important, would be more clearly revealed in these cohort default rates. The vertical axis is the default rate for each cohort-calendar year combination. The most striking aspect of Figure 3. In particular, default rates in and in — were much higher than those experienced by the — cohorts during the recession.

McDonald and Van de Gucht studied aging effects in the context of a duration model. Survival was measured from the time of issuance of the bond. For their sample of high-yield bonds, estimated default intensities tended to increase as the bonds seasoned. McDonald and Van de Gucht attempt to control for supply effects by including covariates that indicate whether the bond was issued in the period — or after This approach, however, may not fully capture the substantial changes in composition of 3.

Statistical Prediction of Bankruptcy Figure 3.

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Default Arrival low-grade debt that occurred in the late s and the associated increases in default rates. Further exploration of these issues seems warranted. Certain step-up corporate bonds have coupon rates linked explicitly to credit rating. Proposed changes to the BIS accord, reviewed in Section 2. Credit derivatives, and other contracts, sometimes have provisions for contingent payments based on changes in credit ratings. For these and related reasons, a model of the risk of ratings changes is a key ingredient for credit-risk management and for the valuation of many creditsensitive instruments.

This chapter reviews some of the historical patterns in ratings transitions as well as alternative approaches to modeling ratings transition risk. Average Transition Frequencies Major ratings agencies report the historical average incidence of transitions among credit ratings and into default in the form of a matrix of average transition frequencies. Each row corresponds to the rating at the beginning of a year; the column heading gives the end-of-year rating. Ratings Transitions Table 4.

The fractions of transitions shown to WR correspond to withdrawn ratings. Although there may be some implications for credit quality for the event of becoming unrated, this effect is often ignored by normalizing each transition frequency by the total fraction of bonds that do not have a withdrawn rating.

It is not uncommon in industry practice to treat an annual average transition frequency matrix of the sort shown in Table 4. Such a Markov-chain assumption for ratings transitions is the basis for the popular CreditMetrics model J. Ratings Risk and the Business Cycle Table 4. One should beware of the potential traps associated with this elegant and simple treatment of the rating of an issuer as a Markov chain.

The reported transition frequencies are only averages and do not condition on all available information. There is indeed apparent momentum in ratings transitions data. Ratings Transitions investment and speculative-grade below Baa ratings categories. These ratios were then correlated with the four-quarter moving average of U. Such a large transition rate is well out of line with typical transition rates.

We report the sample correlations in Table 4. The trends in upgrades relative to downgrades for speculative-grade issues, in particular, are strikingly well correlated with GDP growth during the second half of the sample. In Chapter 3, for example, we reported a consistent large negative correlation between default rates and GDP growth. In order to analyze the impact of business cycles on ratings transitions probabilities, Nickell et al.

European domiciles ; 2 ten industry categories; and 3 the business cycle peak, normal, and trough as explanatory variables. Nickell et al. For U. Ratings Risk and the Business Cycle Figure 4. Ratings Transitions periods. Such a model and estimation procedure implicitly assume independence across issuers within a year, given the business cycle. Correlation is induced only by business cycle changes which affect all issuers the same way. A selection of the results of Nickell et al.

The entries shown in bold type are the percentage transition probabilities estimated for troughs of the business cycle. Those shown in standard type are associated with the peaks. For example, with three business cycle states, as in Nickell et al. Then, in order to deal with multiyear transition probabilities, we can make the additional simplifying assumption that the time series of underlying covariate vectors, X1 , X2 ,.

The unconditional multiyear transition probabilities can be calculated, for example, by simulating the paths of X and averaging the conditional transition probabilities over independently generated paths for X. This is called a doubly stochastic transition model. More computationally tractable doubly stochastic ratings-transition models, in continuous time settings, are introduced in Section 4. Table 4. Suppose, for instance, that Xt has three outcomes, as in the example above with peak, normal, and trough. Multiperiod ratings transition probabilities are now easy, following the matrix-product rule.

Ratings Transitions and Aging There is evidence that the transition rates to new ratings depend on the length of time that an issue has held its current rating and also on an aging effect. These effects are explored by Carty and Fons , who assumed that the probability distribution for the time spent continually in a given credit rating was described by the Weibull duration model introduced in Chapter 3 in the context of default time.

Ratings Transitions Figure 4. Hazard functions for selected long-term ratings. Source: Carty and Fons, Additionally, they examined whether prior ratings changes have predictive power for future ratings changes, that is, whether there is ratings momentum. They found essentially no evidence of a comparable upgrade momentum.

Ordered Probits of Ratings Most of the statistical analyses linking credit ratings to observable creditrelated covariates are special cases of qualitative-response models, such as ordered probit. As opposed to ratings-transition probabilities, these studies focus on an explanation of how ratings are assigned at a given time as a function of currently observable variables related to credit quality. Ordered probit models were estimated by Kaplan and Urwitz , Ederington and Yawitz , Cheung ,2 and Blume et al. Ratings Transitions 4. Ratings as Markov Chains We now extend the notion of rating as a Markov chain by using the intensity approach of Chapter 3, in this case to allow for ratings transition that may occur at any time within a year, as follows.

For example, extrapolating from estimates in work by Jarrow et al. Israel et al. It is well known, however, that for certain 1-year transition matrices, there need not be any generator. Moreover, as pointed out by Israel et al. It is not 3 Israel et al. He found that his econometric model with state-dependent intensities out-performed, in out-of-sample forecasting, various reference models including the constant-intensity model.

This commutativity property obviously holds if the transition intensities are constant. He maintains the special diagonalizability assumption 4. It follows from 4. For example, if the coordinates of X are CIR square-root diffusions, or mean reverting 98 4. We label rating 1 as I , for investment grade, and rating 2 as S , for speculative grade.

We note that this conclusion is independent of assumptions such as 4. In Chapter 6, we reinterpret these results in order to consider the effect of stochastic variation in transition intensities on the term structures of credit spreads for each rating. This chapter introduces some of the most popular of these through the lens of a central building block in most valuation problems—zero-coupon defaultable bonds. In order to highlight key differences among models of default timing, we generally maintain in this chapter the assumption of no recovery by bond investors in the event of default.

In Chapters 6 and 7, we extend the pricing frameworks introduced here to treat corporate and sovereign coupon bonds with nonzero recovery.

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Subsequent chapters examine more complex instruments such as callable and convertible debt and structured products that securitize credit risk in various ways. Both reduced-form and structural models have their proponents in industry and academia. One of the primary objectives of this chapter is to highlight some of the key implications of standard formulations of these models, with particular emphasis on their differences and similarities.

If the issuer is default-free, then valuation can proceed using a conventional default-free term-structure model. Such models are usually based on some short-rate process r , whose stochastic behavior is modeled under risk-neutral probabil 5. Introduction ity assessments.


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There are computational advantages, particularly when working with default, to modeling in a continuous-time setting, for which the analogue to 5. It is not necessary to restrict attention to models that are based on a short-rate process. For example, certain discrete-tenor forward rate models, such as the market model of Miltersen et al. Conceptual Approaches to Valuation of Default Risk If the issuer might default prior to the maturity date T , then, in addition to the risk of changes in r , both the magnitude and timing of the payoff to investors may be uncertain.

Our goal is to characterize this defaultable no-recovery price d 0 t , T under various parameterizations of reduced-form and structural models. This will involve characterizations of the joint distribution of the defaultfree term structure and the default time, based on risk-neutral probabilities. Before going into this basic pricing problem in more depth, we discuss the meaning of risk-neutral probabilities for a setting in which there is the possibility of default. For reasons that we hope to make clear by the conclusion of this chapter, modeling default-risk premia is conceptually more challenging than modeling interest-rate risk.

Risk-Neutral versus Actual Probabilities used.


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The construction of risk-neutral from actual probabilities, or vice versa, will be taken up subsequently when we describe alternative pricing models. The historical default probabilities discussed in Chapter 3 are actual default probabilities and, as such, are not the relevant probabilities for use in pricing. To illustrate the difference between risk-neutral and actual default probabilities, suppose as in Figure 5.

On the other hand, if the issuer defaults before maturity which happens with actual probability 0. Because the default recovery is known with certainty in this example, there is no recovery risk premium. It is natural, especially because default risk is correlated with downturns in the business cycle, to presume that investors demand a premium, above and beyond expected default losses, for bearing default risk.

If so, then our pricing must somehow account for this risk premium. Then, according to the risk-neutral valuation paradigm, the fact that the security is priced at par implies that its price is the present value of the risk-neutral expected payoff, so that Figure 5. Backing out the implied risk-neutral default probability. The superscript asterisk will henceforth be used to indicate risk-neutral variables or properties.

This makes sense if investors are averse to defaulttiming risk. Indeed, one may experience jumps and the other might not. However, this direction is unlikely to be productive when an issuer has little or no actively traded debt or other securities and there are no reasonable proxy security prices.

Examples of both approaches are discussed later in this chapter and in Chapters 6 and 7. Before turning to pricing methods in more depth, it is instructive to informally consider the likely magnitude of risk-neutral default intensities relative to their actual counterparts. That is, how important are defaultrisk premia? In a manner that we analyze in the next chapter, positive recovery upon default narrows spreads.

Thus, on average, credit spreads, such as those shown by generic issuers with B or Ba ratings in Figure 5. Actuarial credit spreads are those implied by assuming that investors are neutral to risk and use historical frequencies of default and average recoveries to estimate default probabilities and expected recoveries, respectively. Figure 5.

For example, the actuarially implied credit spreads on A-rated 5-year U. The corresponding Figure 5. Five-year industrial speculative-grade spreads. Source: Bloomberg. Actuarial credit spreads. Source: Fons, Treasury securities of similar maturities, have been on the order of basis points, as indicated in Table 5. See Chapter 7 for details on these data. The actuarial credit spread of 5-year Ba-rated bonds is about basis points.

The corresponding market yield spreads have been on the order of basis points, as indicated in Figure 5. Differences between actuarial credit spreads and actual yield spreads are due to many effects, including risk premia for bearing default risk, tax shields on treasuries Elton et al. Even after measuring spreads relative to AAA yields thereby stripping out Treasury effects , actuarial credit spreads are smaller than actual market spreads, especially for high-quality bonds. We will bear this in mind as we discuss alternative formulations of defaultable bond pricing models.

An implication of this modeling framework, as shown by Lando , is that the zerorecovery defaultable bond price of 5. Reduced-Form Pricing provided that default has not already occurred by time t. An intuitive interpretation of this pricing relation is as follows. All of these calculations are with respect to risk-neutral probabilities and are based on the assumption of a doubly stochastic default model. That is, X i is, for each i, a CIR also known as a square-root or a Feller diffusion, as introduced in Chapter 3, and these risk factors are risk-neutrally independent.

In certain cases, however, the computational advantages may be worth the approximation error associated with such a Gaussian formulation. In general, s t , T and r t are correlated through their joint dependence on X t. It follows from 5. Reduced-Form Pricing ter 3. As a further illustration, consider the simple mean-reverting-with-jumps model of Section 3. We take a constant default-free short rate r for simplicity. Empirically, as for this simple model, the term structure of yield spreads is downward sloping for low-quality issuers and upward sloping for relatively high-grade issuers, as indicated in Table 5.

More systematic analyses in Jones et al. The shape of the term structure of credit spreads for low-grade bonds has been more controversial. The actuarial yield spreads of Fons illustrated in Figure 5. Term structure of coupon-strip zero-recovery yield spreads. As these authors show, because the spreads of all speculative-grade issuers are averaged, this bias might induce a spurious downward slope to spread curves. By matching bonds by issuer and ratings, Helwege and Turner conclude that spread curves for B-rated U. Subsequently, He et al. From Table 5.

By analogy, the term structure of volatilities of yields on zero-coupon bonds is typically downward sloping provided the short rate r exhibits mean reversion. Short 0. Conceptual Approaches to Valuation of Default Risk 5. Structural Models Structural credit pricing models are based on modeling the stochastic evolution of the balance sheet of the issuer, with default when the issuer is unable or unwilling to meet its obligations.

Default before maturity is not considered. The bond price is thus the initial market value of assets less the initial market value of equity, which is priced using the Black-Scholes formula as though equity is a call option on assets struck at the face value of debt. We consider the price d t , T at time t of a zerocoupon bond of face value D maturing at date T , assuming that no other liabilities mature between t and T , and assuming recovery of all assets in the event of default absolute priority.

Structural Models Geske extended this debt pricing model to the case of bonds maturing at different dates. We suppose for now that the riskless rate is constant at r and that assets are risk-neutrally log-normal, satisfying 5. Based on the analogous result 3. Section 7. Although this approach extends to much richer stochastic environments, its tractability declines rapidly as one enrichs the models used for A and r and allows for a time-varying default threshold D.

Following Zhou , for instance, suppose that 5. With 5. Zhou provides a closed-form expression for the associated risk-neutral survival probability. Boyarchenko and Hilberink and Rogers extend this approach to more general jump-diffusion models. There is a striking difference in the shapes of the spread curves at both short and long maturities.

We address each of these in turn. Credit spreads implied by reduced-form and structural models. Source: Collin-Dufresne and Goldstein, unpublished paper, Comparisons of Model-Implied Spreads default is essentially zero for short horizons. The associated credit spreads are therefore essentially zero for short maturities. This explains the location of the dashed line in Figure 5. In contrast, the solid line for the reduced-form model starts at a nonzero spread and increases gradually, much as we see in the empirical data.

This allows default at jump times, which occur with a positive intensity and therefore cause nontrivial short spreads. Consequently, the term structure of credit spreads takes a shape similar to that of the reduced-form case illustrated in Figure 5. Conceptual Approaches to Valuation of Default Risk These features are shared by many of the commonly used default-intensity models. In assessing the relative merits of structural versus reduced-form models, we should recognize that it is unrealistic to assume, especially at low levels of capital, that balance-sheet data provide a precise view of default risk, particularly over short time horizons.

Accounting data tend to be noisy. An obvious example is the demise of Enron in If one assumes uncertainty about the true asset level, then one obtains a default intensity under natural conditions, and the term structure of default risk and spreads 6 As noted in Chapter 3, forward default probabilities are conceptually analogous to forward interest rates in term-structure models. Dybvig et al. Analogous properties of the forward probabilities follow immediately.

Comparisons of Model-Implied Spreads Figure 5. Credit spreads under perfect and imperfect information. Recovery at default is Conceptual Approaches to Valuation of Default Risk A particularly important aspect of imperfect accounting information is that credit spreads are strictly positive at short maturities, and nontrivially so as compared with the case of perfect accounting information, which would imply as shown zero credit spreads at zero maturity. For the example illustrated in Figure 5. From Actual to Risk-Neutral Intensities Though pricing depends on risk-neutral default probabilities, in many circumstances it would be useful to know the mapping between actual and riskneutral probabilities.

For example, risk-neutral probabilities are not easily inferred from market prices when one is pricing a newly issued security. Conversely, one may wish to use the risk-neutral probabilities implicit in credit spreads as a source of information when estimating actual default probabilities for trading, risk-management, or credit allocation purposes. This section describes some of the mappings used by practitioners that connect actual and risk-neutral probabilities.

In general, the dependence of these intensities on both the state and the likelihood of a 5. From Actual to Risk-Neutral Intensities given path are different under the actual and risk-neutral measures. Here, the 8 Jarrow et al. This would not, however, imply that actual and risk-neutral survival probabilities are the same. A default-risk premium could in this case be due, for example, to an expected growth rate in future default intensities that is risk-neutrally higher than it is in actuality.

As in 3. In order to calibrate this model to market data for the purposes of their internal risk management and credit-derivative pricing systems, J. A similar formulation is applied to the valuation of corporate bonds in Bohn and related studies by KMV Corporation. From 5. This approach does not use market bond price information, but rather infers the model-implied price from historical default information. As part of model validation, after this calibration, one can check whether the market Sharpe ratio implied by the corporate yields and that implied by the model are consistent with what has been observed historically.

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Using a somewhat different model, Huang and Huang have shown that there is some degree of inconsistency between the risk premium on corporate debt and that on equity, which might be interpreted as a source of information regarding the liquidity component of spreads on bonds. In more general settings, the distributional assumptions underlying the calculation 5.

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Nevertheless, this approach provides one tractable and economically motivated approximate mapping between actual and risk-neutral survival probabilities. Focusing initially on corporate debt, we begin by extending the basic reduced-form models introduced in Chapter 5 to allow for nonzero recovery and for potentially different liquidities between the corporate and reference bonds over which corporate yields are spread. We then turn to conventional structural models of corporate bond prices. This is followed by a discussion of some extensions of our pricing framework that accommodate ratings-transition risk.

Finally, we address several new issues that arise in pricing sovereign debt, including the need to allow for multiple types of credit events. Uncertain Recovery A variety of recovery assumptions appears in the bond pricing literature, with sometimes material differences in tractability and pricing behavior. Reduced-form models used for term-structure modeling, including creditderivative pricing among other applications, are distinguished mainly by their treatments of recovery. In their simplest forms, all assume that, conditional on the arrival of default in the next instant, the bond in question has a given expected fractional recovery.

But this recovery is a fraction of what? One class of models, introduced by Jarrow and Turnbull , has taken recovery at default to be a given fraction of a default-free but otherwise equivalent bond. Finally, there is a class of models that takes recovery to be a 6. In principle, one would like to discount the eventual actual payouts to bondholders, net of procedural costs, back to the date of default. However, this information is rarely available. Carey presents recovery results on this basis for private debt. This proxy is likely to be imperfect, but, as it is based on market prices, it is in principle the value that bondholders would receive by selling their positions following default.

Figure 6. Median recovery rates for bank loans and equipment trusts were higher than those for all rated, publicly issued corporate debt in the United States. Median recovery rates for bonds naturally decline with seniority. For junior subordinated bonds, the median recovery rate was only about 10 cents on the dollar. The rectangles in Figure 6. Distributions of recovery by seniority. Pricing Corporate and Sovereign Bonds p 75th percentiles of the recovery distributions.

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