Quantifying credit risk II: Debt valuation



Quantifying credit risk II: Debt valuation

Stephen Kealhofer

9,258 words

1 May 2003

Financial Analysts Journal

78-92,15

Volume 59, Issue 3; ISSN: 0015-198X

English

Copyright (c) 2003 ProQuest Information and Learning. All rights reserved. Copyright Association for Investment Management and Research May/Jun 2003

"Quantifying Credit Risk I" (in the January/February 2003 Financial Analysts Journal) presented evidence on the effectiveness of using the information in market equity prices to predict corporate default. This Part II links those results to the valuation of corporate debt and shows that, contrary to previous negative results, the approach pioneered by Fischer Black, Myron Schales, and Robert Merton provides superior explanations of secondary-market debt prices.

A major frontier in modern finance is the quantification of credit risk. More than 25 years ago, Black and Scholes (1973) proposed that one could view the equity of a company as a call option. This insight provided a coherent framework for the objective measurement of credit risk. As subsequently elaborated by Merton (1973, 1974), Black and Cox (1976), and Ingersoll (1977a), this approach has come to be called the Merton model.

Initially, empirical research based on the Merton model produced discouraging results,1 and even today, new researchers continue to confirm these original negative results.2 In 1984, however, Vasicek (in an early form of what ultimately became known as the KMV model) took a novel approach to implementing the Merton model that has had considerable success in measuring credit risk. The KMV version of the Merton model, which has been extended over the years, has become a de facto standard for default-risk measurement in the world of credit risk.3

Parts I and II of "Quantifying Credit Risk" are intended to be a compendium of the empirical research on the KMV model over approximately the 1989-2001 period. Part I addressed the differences in implementation between the KMV model and the Merton model and reviewed studies comparing the performance of the KMV model with the performance of agency debt ratings in predicting default.4 This Part II presents the results from studies of using the model to value corporate debt relative to the results for using alternative approaches.

The thesis of the two-part "Quantifying Credit Risk" is that the Black-Scholes-Merton approach, appropriately executed, provides the long-sought quantification of credit risk. It is an objective, cause-and-effect model that provides analytical insight into corporate behavior. Driven by information in a company's public equity price, the model produces empirical estimates of default probability that outperform other well-accepted public benchmarks, such as agency debt ratings, in predicting default. Finally, in contrast to the earlier findings, the Merton-based KMV model produces results that explain 60-70 percent of the cross-sectional variation in secondary-market corporate bond spreads and that fit the secondary-market values of corporate bonds better than results produced by conventional matrix pricing approaches using agency bond ratings.

The KMV Model and Bond Prices

The ultimate aim of default-risk prediction is to value liabilities subject to default. Decisions involving credit risk reduce to propositions about the desirability of transacting a given instrument or entering into a particular contractual arrangement. These decisions are value decisions; that is, given the default risk, some price will make the transaction attractive, some price will make it unattractive.

The general impression in the academic community has been that the Merton approach does not work well empirically in valuing liabilities subject to default. This view was summarized by Kao (2000), who wrote:

Empirically, most criticisms of Merton's basic model center on its difficulty in explaining the observed term premium of the corporate bond yield curve. . . . Jones, Mason, and Rosenfeld (1984) showed that the basic model is not successful in pricing investment-grade corporate bonds, even for those issuers with a simple capital structure. (p. 59)

Recent empirical work has begun to challenge these conclusions but has not yet documented the source of the difficulties encountered by such researchers as Jones, Mason, and Rosenfeld (henceforth, JMR).5 Logically, the previous implementations of Merton could have failed either because the default rates they implied were empirically false or because the option framework provided an incorrect link between default probabilities and actual debt prices. This incorrect link, in turn, could have failed because debt is irrationally priced (given its default risk), the options framework is incorrect, or applying the framework correctly to actual companies is not feasible for computational or data reasons.

The strategy of the KMV model is to measure the default rates correctly. Thus, if default rates are the source of the Merton model problems, then obtaining better default rates should provide a resolution of the valuation problems.

Predicting Default Rates

The accepted facts about bond yields that a model for valuing corporate bonds must deal with are as follows:

* Spread volatility. The average yield spread associated with a given agency rating grade changes considerably over time. For an example, see Figure 1.

* Considerable variation in the shape of spread curves by rating grade as a function of term. This variation is illustrated in Figure 2, which shows the average shape of spread curves by rating grade as a function of term. It was obtained by calculating yields to worst call on a monthly basis for U.S corporate bonds, expressing them as spreads over U.S. Treasury securities, and averaging them over the period January 1992 through January 1996. As a rule, investment-grade spreads slope upward as terms increase; BBB spreads tend to be flat or slightly humped; and CCC spreads tend to be downward sloping.

Any model that purports to value corporate bonds has to address these issues. If one is using debt ratings as a measure of default risk, then one has to introduce at least one additional variable to explain spread widening and narrowing. If the shape of the spread/term curves is variable and complex, it will be difficult to explain without introducing a whole set of new variables. Figure 1. Median Expected Default Frequencies and Spreads for BBB Rated Debt, September 1995-September 2000 Figure 2. Term Structure of Credit Spreads, Monthly Data, 1992-January 1996

These problems are largely mitigated by using the KMV expected default frequency (EDF(TM)) credit measure as the basis of the default-risk measurement. First, EDFs indicate that the typical default probability associated with an agency rating grade changes considerably through time and substantially explains observed spread widening and narrowing. Second, EDFs themselves have a term structure that substantially explains the observed spread term structures.

Accurate valuation requires accurate absolute default rates. Part I of "Quantifying Credit Risk" presented results on the default-predictive power of the KMV model, but tests of the default-predictive power of the KMV model and the EDFs use only the rankings implied by the model. The rankings are based on the distance to default (i.e., the number of standard deviations to the default point). These distances are an output of the model and are not statistically fitted to historical data. In other words, given the asset value, volatility, and default point, one can calculate the distance to default; no fitted coefficients or statistical estimates are required.

The absolute levels of the EDFs, however, are calibrated to historical experience. Thus, one needs to separately examine whether the actual levels of the EDFs are accurate as ex ante predictors of default rates. To obtain reliable results, this test should be done on out-of-sample data because the function between distance-to-default values and EDFs is empirically fitted.

The following study examined this issue (see Riley 1992). The sample was U.S. public nonfinancial corporations with EDFs available in the period January 1989 (the default-rate mapping in use by KMV at that time had been fitted only through 1988) through December 1991. Cohorts were formed by putting the companies into five groups of approximately equal size according to their EDF values as of the start of the period. The default experience of each cohort was then tracked until the end of the study, with cumulative default rates recorded at six-month intervals. New cohorts were formed every six months, and the results were averaged across the cohorts of the same quality for fixed horizons. Companies that disappeared for reasons other than default were tabulated as a separate category of outcome.

If EDFs correctly measure predicted default rates, the average EDF of a cohort should be close to its subsequent realized default rate. With cohorts starting every six months, six observations were available for six-month default rates, five observations for one-year default rates, and ultimately, only one observation for three-year default rates. The results from the study are summarized in Panel A of Figure 3, which shows that the correspondence between the predicted and the actual default rates was good overall, somewhat better for the higher-quality companies, and somewhat worse for the lower-quality companies.

Gurnaney (1998) made a subsequent study using 1995-97 data and following the same procedures. This study used a later version of the model that had been fitted to empirical default rates on the basis of 1978-94 data. These results are summarized in Panel B of Figure 3. They show a striking relationship between the predicted and the actual default rates.6

An extension was made, as part of the Gurnaney study, in which the average credit quality of each original cohort was followed over a period of six years, remeasured every year, and compared with subsequent realized default experience. As one would expect, on a year-by-year basis, the ex ante versus ex post default rates varied considerably because of prediction error and contemporaneously correlated events of default. Figure 4 depicts realized default rates versus average EDFs for cohorts formed in 1990 and remeasured every year through 1996. Note that the cohorts represent different original credit-quality groups. (They are depicted on two graphs rather than one because of the scale differences between the cohorts; Cohort 3 is shown on both for comparison.) Figure 4 shows, however, definite dynamic tracking of ex post default rates by the ex ante predicted default rates.

A key feature of these data is the pattern of subsequent default rates associated with a given credit-quality cohort. Even when a single cohort is used, one can see the fall in average default rate . associated with initial low-quality groups versus the increase in average default rate associated with initial high-quality groups. This behavior is what induces the observed structure of default term premiums.

To accurately determine default term premiums, one can empirically assign expected default rates to the distances to default one has determined for different horizons. EDFs are determined for one-year through five-year horizons. If one averages over all companies with similar one-year EDFs and then looks at the average annualized default rates for different horizons as a function of the original EDF, this same pattern emerges, as is shown in Panel A of Figure 5.

The pattern of expected default rate versus time horizon as a function of initial credit quality was theoretically predicted by Merton in 1974 as a consequence of the Merton model. The Merton results, as subsequently corrected by Pitts and Selby (1983), are shown in Panel B of Figure 5. The empirical evidence confirms this prediction qualitatively, although the exact shapes of the curves hypothesized by Merton do not correspond to the (flatter) observed shapes from empirical studies.7 By taking the output from a Merton model to be the number of standard deviations to the default point for a given horizon for a given company, treating it as ordinal, and fitting these data to empirical default rates, the KMV approach potentially addresses this problem. Figure 3. Predicted and Actual Default Rates

In summary, changes occur through time in the average level of default risk within the population at large as well as in subgroups of the population. These changes can be detected in advance by using the KMV model and are consistent with subsequent actual default experience. The underlying point is that because EDFs incorporate equity value information, they are able to dynamically predict company default rates. Figure 4. Predicted EDFs versus Actual Default Rates for Cohorts Formed in 1990

Furthermore, the cumulative default experience of a given cohort of companies has a time pattern that depends on the cohort's initial credit quality. The characteristics of that pattern are consistent with the general Merton approach and are incorporated in the KMV model via empirically fitting default rates for different future time horizons to the model's output, namely, the standardized distance to default for the respective horizon. Figure 5. Credit Spreads for Groups with Different Initial Credit Quality as a Function of Term

The practical import of changes in default rates for bond valuation can be seen in the typical EDF associated with companies rated BBB by Standard & Poor's over a five-year interval. A look again at Figure 1 shows that the differences over time are as large as a factor of four. (Figure 1 also shows median BBB spreads to LIBOR and Treasury rates.) As noted previously, the evidence is that these differences correspond to subsequent realized default-rate differences. So, one would certainly expect to see them reflected in the pricing of corporate debt, which could then explain the spread volatility associated with a given agency rating class.

The practical import of the default-rate term structures for valuation can be seen by comparing the shape of the EDF term structures in Panel A of Figure 5 with the term structure of the credit spread in Figure 2. These two pictures have strong qualitative similarities, which suggests that the term structure of expected default rates may constitute a significant explanation for the complex shape of observed credit-spread term structures.

To develop these implications more explicitly requires a review of the relationship between expected default probabilities and the pricing of credit instruments, such as bonds.

Default Rates and Firm Values

From option-pricing theory, one can obtain a simple characterization of the relationship between the default probability for a company and the valuation of its liabilities. Consider the case of a single cash flow, F, due at a single future date, t. Let r be the continuous discount rate to t for a default-risk-free cash flow. Then, an option theoretic formula for the value of the cash flow today, V, is8

V = Fexp[-r^sub t^][1 - q^sub t^(LGD)], (1)

where q^sub t^ is the so-called risk-neutral cumulative default probability to t and LGD is the "loss given default" term (that is, the expected percentage loss should the borrower default).9 The relationship between q^sub t^ (the risk-neutral cumulative default probability) and p^sub t^ (the actual cumulative default probability to t) under the lognormality assumption is given by10

where

N = standard cumulative normal distribution

N^sup -1^ = inverse function of standard cumulative normal distribution

[mu]^sub A^ = instantaneous expected return to the asset

[sigma]^sub A^ = volatility of asset returns

The cash flow is valued as if the default probability were q^sub t^, which is greater than actual probability p^sub t^. The probability used for pricing purposes, the risk-neutral probability, can be converted into the actual default probability and vice versa with knowledge of the underlying asset's required return. The difference between the actual probability and the pricing probability stems from the systematic component of default risk.11 The greater is the systematic component of this risk, the larger is the difference between q^sub t^ and p^sub t^.

Mathematically, the difference between p^sub t^ and q^sub t^ is determined from the expected return that is required for the systematic risk of the underlying asset. If the underlying asset had no systematic risk (so its expected return was equal to the risk-free return), the two probabilities would be identical. Because the systematic risk of the underlying asset is communicated to the cash flow via default, the amount of risk premium required for credit-pricing purposes is directly related to the amount of risk premium required for the underlying asset.

Equation 1 assumes a single cash flow at a single date. Generalizing to multiple cash flows at multiple dates introduces significant complexity. If one assumes a relatively straightforward perpetuity model with an absorbing barrier, however, one can show that the single-cash-flow, single-date model understates the absorption probability by approximately a factor of 2. This aspect yields a restatement of Equation 2 in which p^sub t^ is consistent with the empirical definition of the EDF as being the absorption probability, or cumulative probability of default, to date t:12

where

r^sub A^ = asset return

r^sub M^ = market return

[mu]^sub M^= expected return

[sigma]^sub A^ = standard deviation of asset return

[sigma]^sub M^ = standard deviation of market return

[rho] = correlation of r^sub A^ and r^sub M^

[lambda] = market Sharpe ratio

When terms are substituted in Equation 3, the risk-neutral probability can be written as

The risk premium on the systematic risk divided by its standard deviation is, as noted, the Sharpe ratio of the market, or equivalently, the market price of risk. If the equity market is the benchmark, the expectation is a value for the numerator of 0.06-0.08 and for the denominator of 0.12-0.20. These values suggest a value for the market price of risk in the range of 0.3-0.7. There is no reason to believe this value cannot vary through time, however, and explain some of the variation that one observes in spreads on debt.

An extension of the valuation formula for a single cash flow to the case of multiple cash flows is13

v = [Sigma]C^sub t^exp[-r^sub t^t][1 - q^sub t^LGD], (7)

where C^sub t^ is the cash flow at time t and the summation is over all cash flow dates.

A key requirement for using this specification is a measure of loss given default. Although LGD has been the subject of recent empirical work, much about it remains unknown. For the valuation work I will describe in the next section, LGD values were determined by security and subordination and were set to the average historical values of 0.42 for senior secured, 0.52 for senior unsecured, 0.66 for senior subordinated, and 0.69 for junior subordinated debt.14 One implication of this choice is that the estimated value of lambda, the market price of risk, is identified relative only to LGD. If the chosen values are too low, the effect is to inflate the lambda estimate, and vice versa.

Another problem arises if LGD has cross-sectional variation that correlates with default probability; this characteristic makes it difficult to separately identify the effect of default probability. Finally, by ignoring variation in LGD over time, one might infer that the market price of risk is changing when, in fact, recovery amounts are changing. In essence, this choice of specification makes the default probability do all the work in fitting the bond prices.15

Corporate Bond Valuation

As previously noted, the empirical evidence on the Merton approach for valuing corporate bonds has been discouraging. The best-known work, that of JMR, found that implementation of the Merton approach had some explanatory power for below-investment-grade bond prices, but overall, it performed little better than a model that assumed a constant spread over risk-free pricing. Ogden (1987) produced similar results in a later study.

This article extends work by Bohn (2000a) that used data on 24,465 bond issues representing 1,749 issuing companies over the period June 1992 through December 1998. Using EDFs from the KMV implementation of Merton and the procedures described here, Bohn obtained good fits to bond spreads through fitting only the market Sharpe ratio.

The values estimated by Bohn for the market Sharpe ratio, shown in Figure 6, range approximately between 0.3 and 0.5, which is quite consistent with reasonable equity market estimates. The estimates display little trend, and some of the observed variation in the estimates is certainly a result of sampling error. This result has an important implication: The large downward move in spreads over this period of time resulted mostly from decreases in default probabilities rather than changes in the required risk premium on corporate bonds. Figure 6. Estimated Market Sharpe Ratio, June 1992-December 1998

Note also, however, that the evidence from the Bohrt study on the constancy of lambda, the debt market risk premium, presents difficulties. The estimation procedure for EDFs assumes a time-invariant drift rate for underlying company asset values. Thus, any marketwide variation in the risk premium in the equity market would show up in the EDFs as a change in expected default probability. This variation would not necessarily be readily discernible in the out-of-sample default-level predictions because if the equity risk premium tends to return to a normal level over time, the variation will look like sampling error.

Keeping this caveat in mind, note nevertheless the clear time variation in realized default rates, which is consistent with the aggregate movement of EDFs from the KMV model, as illustrated in Figure 4. This picture is very different from the picture one obtains when looking at spreads through the lens of agency debt ratings. For a given rating range, considerable movement in spreads over time is evident. Under the implicit assumption that the typical default rate for BBBs does not change much over time, one could thus mistakenly interpret this movement as substantial variation in bond risk premiums. Figure 1 gives the median spread over Treasuries (and over LIBOR) for BBB rated, publicly traded bonds from September 1995 to September 2000 versus the median EDF for BBB rated, publicly traded, nonfinancial companies over the same period. Based on a population of 230-350 issuers in 1994, the median EDF was found to be 0.25 percent; in 1997, it was 0.10 percent; in 1999, it was 0.50 percent. These types of dramatic moves in credit quality are typical, and they explain much of the movement in BBB spreads over this same period. Similar patterns in spreads can be documented for every credit-quality range.

Bohn's results do not directly address the negative findings of the JMR paper, however, in the sense of documenting the performance of the Merton approach versus an alternative in attempting to explain credit spreads. This article extends Bohn's research by examining a number of alternative specifications of the credit spread and comparing their performance with that of the KMV model.

The simple alternative model used by JMR was to assume a single constant credit spread. The finding that a constant credit spread had approximately the same explanatory power as their implementation of Merton was at the heart of the negative JMR findings. Any specification that produces a statistically and economically significant explanation of credit spreads will do better than the JMR null hypothesis, so it does not provide a strong enough alternative for evaluating the KMV approach. The strongest alternative to the Merton approach for explaining credit spreads is the best available alternative measure of credit risk, namely, agency ratings.

Two specifications are tested here. The first assumes a single credit spread per agency rating grade at any given moment; that is, it assumes a single annual risk-neutral pricing probability (or EDF) in the seven-parameter model, q'^sub t^, for each rating grade. This type of model lies at the heart of most matrix pricing models of credit spreads. It is expressed mathematically as

V = [Sigma]C^sub t^exp[-r^sub t^t][1 - (q'^sub i,t^t)LGD], (8)

with i = 1,..., 7 by rating grade. For ease of reference, I will call this model the "seven-parameter model."

Another specification (with fewer parameters) is to use the annualized probability of default by rating grade and by term derived from the reported cumulative default rates of the given rating grade, based on the rating agency's own estimates. This model is directly comparable to the Merton approach because it requires an estimate of only a single pricing parameter across all the grades and terms. Mathematically, with q'' the risk-neutral pricing probability (or EDF) in the single-parameter model, it is

q''^sub i,t^ = N[N^sup -1^(p^sub i,t^) + [lambda](t^sup 1/2^)], (9)

where p^sub i,t^ is the cumulative probability of default for rating grade i and maturity t and [lambda] is the estimated pricing parameter. For ease of reference, I will call this model the "single-parameter model."

In addition to these alternatives, two other important differences with the JMR study significantly strengthen the power and focus of the study reported here. First, the dependent variable I used is the bond credit spread, rather than the bond yield. Much of the explained variation in the JMR study arose from variation in the risk-free term structure. Because none of the approaches described so far attempts to explain variation in the risk-free term structure, using the total yield rather than the credit spread as the dependent variable obscures the explanation of the credit spread.

The determination of the credit spread is itself a controversial issue. Isolating the credit spread presents two main difficulties. The first is that the appropriate risk-free base rate is not given by the Treasury term structure but is quite close to the LIBOR (swap rate) term structure (see, for example, Collin-Dufresne and Solnik 2001). The second point is that most corporate bonds are callable, meaning that some adjustment needs to be made to their pricing to reflect the value of the call option that the investor has given up. These points are both addressed at length in Bohn (2000a). For current purposes, a procedure was used that calculates a risk-free term structure that is very close to the LIBOR term structure; no material difference in results would have resulted from using the LIBOR term structure exactly. Callability (and other optionality) is addressed by using standard option-adjustment valuation methods.17 For measures of term, option-adjusted duration was used for consistency. The bonds were treated as if they were zero-coupon instruments with maturities equal to their (option-adjusted) durations.18

The second major difference between the study reported here and the JMR study is the amount of data used. The data for this study came from EJV, a subsidiary of Reuters, and cover the period January through September 2001. Data sets of bond prices contain many observations on bonds that have not traded in recent memory. To remove these made-up prices, one would ideally use information on trading volume to identify nontraded bonds, but such data are not currently available. As an alternative, I used a quarterly data set of bond trades (to eliminate any bonds that were not traded at least once within the quarter) based on trades of insurance company portfolios. This process eliminated approximately 64 percent of the observations.

In addition to filtering out bonds that apparently had not traded, I eliminated bonds with the following characteristics: bonds with optionality other than callability (convertibles and putables), bonds near call, pay-in-kind bonds (PIKs), bonds issued by real estate investment trusts (REITs), small issues (less than $10 million), issues maturing in less than a year or more than 10 years, and bonds with option-adjusted spreads greater than 20 percent. These exclusions were intended to eliminate bonds whose quotes were likely to be wrong. For instance, bonds with little time remaining to maturity or small issue amounts tend not to trade. Most, but not all, of these bonds were eliminated by the requirement that the bond must have traded recently. The results reported here are not sensitive to these exclusions, but the exclusions generally helped to clean up the problems caused by the absence of actual transaction prices.

The resulting data set had an average of 2,199.3 bonds on 9 dates.20 This study used nearly 100 times the data of JMR (27 bonds on an average of 11.3 dates each) and Ogden (57 bonds on 1 date each, namely, the bond's issue date).

The overall results of this study are summarized in Figure 7, which provides the R^sup 2^ estimates for the KMV variant of the Merton approach and two alternative models from each daily cross-sectional estimation over the entire sample period. The significant result is that the Merton approach consistently explains the cross-sectional variation in credit spreads better than either of the two specifications based on debt ratings.

Figure 8 provides the estimates of lambda over time produced by the KMV model (Panel A), the equivalent lambda estimate for the default probabilities based on rating grade from the one-parameter model (Panel B), and the estimates from the seven-parameter model (Panel C). All of these estimates are reasonably well behaved. The estimated risk-neutral probabilities by bond rating grade (Panel B) are generally ordered as per the grades. The results for the ratings-based approaches support the views that (1) the term structure of default probabilities is useful in fitting bond spreads and (2) the difference in default rates between rating grades is stable in relative terms.21 Figure 7. R^sup 2^ of KMV Model Values vs. Alternatives, January-September 2001

Overall, these results lead to a conclusion completely contrary to the original JMR finding. Whereas JMR concluded that the Merton approach had little marginal explanatory power for credit spreads, in fact, as much as 70 percent of the cross-sectional variation in credit spreads can be explained by the KMV variant of the Merton approach. Moreover, this approach not only yields a high proportion of explained variation, but it also beats alternative models of the credit spread based on the best available alternative measure of credit risk-agency ratings.

These results raise serious questions about the source of the original JMR results. First, remember the dramatic difference in the size of the two studies. The JMR results were based on only 27 bonds on an average of 11.3 dates each, whereas the current study used an average of 2,199.3 bonds on 9 dates each. The key observation about the JMR implementation of Merton versus the KMV implementation, however, is that JMR used the assumption of lognormality to derive the (implicit) default probabilities whereas the KMV model uses the actual empirical distribution of default. The two approaches have significant economic differences.

To understand the JMR results, note that most investment-grade companies are more than four standard deviations from default. Under the normal distribution, this situation leads to calculated credit spreads that are essentially zero. For the KMV model, it leads to economically relevant credit spreads in the neighborhood of 4-116 bps. The credit spreads under the lognormal case versus the KMV approach are shown in Figure 9, as a function of distance to default. Figure 8. Lambda Estimates for Implied Spreads, January-September 2001 Figure 9. Lognormal versus Empirical Risk-Neutral Default Probabilities

As a practical matter, in the JMR analysis of investment-grade bonds, little difference was observed between the alternative model, which assumed no credit risk, and their Merton model, which generated few nonzero credit spreads. In fact, their model did yield some explanatory power for below-investment-grade bonds, but it was partially obscured by their attempt to explain the total variation in the bond yield instead of focusing on the credit spread alone. The spreads of below-investment-grade bonds based on the lognormality assumption tend to be considerably more variable than those generated by the KMV model. Many of these companies are more than three standard deviations from the default point, with the result that their spreads are small or zero; companies less than two standard deviations away from the default point tend to have large credit spreads. The largest portion of empirical credit spreads lies in a range that is generated by the conventional Merton approach only if the company happens to lie between two and three standard deviations from the default point. In contrast, the KMV approach associates these same spreads with companies that lie between one and ten standard deviations from the default point. In summary, the conventional lognormality assumptions imply a narrow critical region for generating empirically relevant credit spreads.

The ability of the KMV model to fit credit spreads with a single parameter is also a surprising result that needs further explanation. For example, in the study I have described, some of the alternative specifications based on rating grades used up to seven fitted parameters, but they still did not do as well as the KMV model. The difference stems from two features. First, the correspondence between levels of the individual companies' default probabilities based on EDFs and their credit spreads is good.22 Second, the term structure of EDFs by credit-quality range mimics the spread patterns. In other words, the term structure of EDFs, by correctly describing the average default rate over time, provides a good match to the pattern of empirical spread data.23 Thus, fitting the spreads was largely a matter of finding the right value of the market Sharpe ratio, which acted as a scaling parameter.

These results can be illustrated by using two bonds issued by Verizon Communications, which are presented in Figure 10. The solid line is the spread of EDF-implied spreads (EIS), as described previously, over LIBOR. The dotted line is the spread over LIBOR of the actual secondary-market bond, derived from option-adjusted spreads (OAS) reported by EJV. The dots are constant-maturity quotes (CDS), reported as the midpoint of bid-ask. They are vertically arrayed because they are for maturities of one, three, and five years.

In the case of both bonds (the one with a maturity of April 2008 in Panel A and the one with a maturity of March 2004 in Panel B), the correspondence between the reported secondary-market spreads and the model spreads (the two lines) is close. Careful examination of the data reveals a tendency for the reported bond spreads to make large one-day jumps, however, whereas the EDF-implied spreads evolve more connectedly.

This example illustrates two properties that were discussed previously. First, there is a close contemporaneous relationship between the levels of the EDF, and thus of the EDF-implied spreads, for a particular company and the spreads on the company's bonds. Second, the term differences in EDFs, and thus in the EDF-implied spreads, do a good job of explaining the term differences in spreads, which is illustrated here by the matching of spreads for two bonds of different maturities.

Finally, keep in mind that the model is not fit to this particular issuer or the issuer's bonds. Only the single, marketwide Sharpe ratio parameter, lambda, has been fit to the overall set of bond spreads. Even in this simple example, one can readily see that the theoretical relationship postulated by Black, Scholes, and Merton between a company's equity and liability values has significant empirical validity.

Casual examination of many graphs that, like Figure 10, contrast the EDF and bond spreads of a particular company suggests that changes in EDFs are coincident with or tend to lead changes in market bond spreads. The relationship is not invariant, however, and in some situations, changes in bond spreads lead changes in EDFs. The difficulty in interpreting these graphs is how to decide for companies that do not default whether any given move in EDFs or spreads represents a change consistent with underlying credit quality or is simply noise that will tend to reverse itself over time.

To explore the issue of noise versus signal in EDF moves, Vasicek (1995) conducted a study of spreads versus EDFs. Using a simple approach, he determined which bonds appeared to be underpriced or overpriced relative to their EDFs.24 If the information contained in EDFs, and not contained in spreads, was simply noise, there should have been no return differential between the portfolios. If the EDFs contained information not present in the spread data, then over time, one would expect the spreads to ultimately move toward the values implied by the EDFs, generating a differential return for the undervalued vis-a-vis the overvalued portfolio.

The results of this study are graphed in Figure 11, with the average annual returns provided in Table 1. In fact, the return difference between the under- and overvalued portfolios is a hefty 4.04 percentage points a year.25 This result is consistent with findings that reported bond pricing does not necessarily reflect actual trades but, rather, dealer indications or model prices. It does demonstrate, however, that EDFs contain information that is not contained in reported bond spreads. The fact that the KMV model does not perfectly fit bond spreads is thus explainable, at least in part, by "noisy" bond prices.

Conclusion

Over the past decade, the probabilities of default determined from the KMV model, expected default frequencies or EDFs, have become a major tool of credit-risk transactors and portfolio managers. From the descriptions in this article of the performance of the Merton-based KMV model, the following conclusions are warranted:

* A strong link exists between the credit-risk component of corporate bond prices and the equity price for the same company; this link works much the way initially hypothesized by Black, Scholes, and Merton. Figure 10. Actual vs. EDF-Implied Spreads for Two Verizon Bonds, January 1999-March 2002

* The KMV implementation of Merton theory does a consistently better job of explaining individual corporate bond prices than do conventional alternatives based on agency debt ratings.

* Because of the correspondence between EDFs and actual default rates, the relationship between bond price behavior and the behavior of EDFs is good. Differences between reported bond prices and KMV model prices are at least partially a result of noise in reported bond prices.

* The KMV implementation explains much of the movement in corporate bond prices on an aggregate level after the risk-free term structure has been accounted for.

* The variation in bond spreads over time appears to be mostly a result of variations in default probabilities, not variations in the market price of risk; the pricing of risk in the equity markets is comparable with pricing of risk in the credit markets after the option nature of credit risk has been accounted for. Table 1. Average Annual Portfolio Returns, December 1990-January 1995 Figure 11. Performance of Underpriced and Overpriced Bond Portfolios, December 1990-January 1995

These results are significant for practitioners. The approach to credit developed by Black, Scholes, Merton, and others has long provided a coherent theoretical framework for corporate finance by linking together the asset, equity, and liability behaviors of the company. Its acceptance, however, has always been clouded by the lack of positive empirical results on corporate debt valuation. As a consequence, a variety of ad hoc approaches to credit have proliferated and considerable confusion reigns concerning the "right" way to look at credit. The results I have reported here should help dispel the cloud that has hung over the Merton approach and bolster the approach as the predominant paradigm, theoretical and empirical, for understanding credit risk and corporate finance in general.

Finally, note that the valuation results have all been obtained with relatively simple assumptions about loss in the event of default. Considerable opportunity to obtain an even better understanding and better fits to bond prices remains through further work on the relationships among bond covenants, company characteristics, capital structure, and recovery values in default.

Stephen Kealhofer is a managing director at KMV Corporation, San Francisco.

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Prior research using the Merton approach to value corporate liabilities has yielded negative results. This article reports on new research indicating that the Merton approach, in fact, yields quite accurate estimates of the secondary-market prices of corporate bonds. The primary difference between this research and earlier work is the substitution of empirically based default probabilities for the lognormality assumption of the conventional Merton method.

This article is the second in a two-part series on quantifying credit risk. Part I, which was published in the January/February 2003 issue of the Financial Analysts Journal, presented results of research on the model's default-predictive power. The ultimate aim of default prediction is to determine the appropriate market value of default risk. Part II explores the connection between default prediction and valuation. First, it shows that the levels of predicted individual and aggregate probabilities of default from the KMV model display considerable variation through time and that these predictions correspond to the actual, subsequently realized levels of default. Second, contrary to past understanding, most of the variation in the spreads on corporate bonds can be attributed to variation in expected default probability, not to the risk premium on corporate bonds, which is relatively stable through time. Third, Part II shows that the term structure of bond spreads corresponds empirically to the term structure of default risk.

Finally, the default probabilities from the KMV model are used in an option theoretic framework to explain individual corporate debt spreads. In sharp contrast to previous academic work, the tests I explore found that the generalized Merton approach provides significantly better explanations of debt prices than those provided by alternative models. These tests used significantly more data and more powerful alternative models than previous studies.

The findings suggest that the Black-Scholes-Merton approach, when appropriately executed, provides the long-sought quantification of credit risk. As an objective cause-and-effect model, the KMV model also provides analytical insights into corporate behavior, thus creating the basis for a continuing rich research program into default risk, capital structure, and debt valuation. For portfolio managers, the link from debt values to equity values featured in the KMV model provides a way to understand the correlations between individual credit risks and the correlation between the performance of equity and corporate debt portfolios. For managers of debt portfolios, the results suggest significant new approaches to decomposing and evaluating portfolio performance.

Keywords: Debt Investments: credit analysis; Portfolio Management: debt strategies

Footnotes:

1. Notably, Jones, Mason, and Rosenfeld (1984) and Ogden (1987).

2. See Jarrow and Van Deventer (1999).

3. For instance, Moody's Investors Service has recently decided to use a Merton approach to predict default (see Cass 2000).

4. For a description of theoretical issues underlying implementation of the KMV model, see Crouhy, Galai, and Mark (2001).

5. Empirical works challenging the conclusion that the Merton approach does not work well for valuing liabilities subject to default include Delianedis and Geske (1998), Eom, Helwege, and Huang (2000), and Wei and Guo (1997).

6. The statistical significance of these results is difficult to characterize exactly. If a common factor is affecting firm values, the default rate for any single company will be a nonlinear function of this systematic factor as well as any idiosyncratic effect. The aggregate default rate for that cohort will then reflect a single observation on the aggregate factor but will reflect a sample size equal to the number of companies on the idiosyncratic effects. Thus, to the extent that the observations in the particular period are mostly characterized by aggregate effects, the effective sample size is small. The earlier sample period included a major recession; the subsequent sample period had no major aggregate events.

7. This correspondence was first noted by Sarig and Warga (1989) based on bond yields.

8. Many models yield this or similar formulas (see Bohn 2000b). A simple intuitive derivation is to consider the Black-Scholes analog. If the equity, E, is seen as the call option and if there are just two claims, equity and debt, D, against the value of the underlying asset (firm value), A, then the debt value is given by A - E; that is, D = (1 - N^sub j^)A + Fexp[-r^sub t^](1 - N^sub 2^), where N^sub 1^ and N^sub 2^ are the standard Black-Scholes coefficients. The first term can be interpreted as the value of the recovery in the event of default, and the second term, as the value of the bond assuming no recovery. The coefficient N^sub 2^, the risk-neutral default probability, is equal to N(Z^sub 2^), which is the risk-neutral distance to default. Equation 1 then follows from expressing the recovery via the LGD term.

9. As Note 8 indicates, this specification requires a value for the LGD, whereas in the canonical Merton approach, the recovery amount is endogenous.

10. EDFs are quoted as annualized default probabilities. The cumulative default probability is determined by compounding the annualized survival rate and subtracting it from 1. In other words, p^sub t^ = 1 - [1 - EDF^sub t^]^sup t^.

11. For related discussions of actual versus risk-neutral probabilities, see Demchak (2000) and Crouhy et al.

12. This result is from Vasicek (2001). The quality of the approximation breaks down if p^sub t^ is too large.

13. Equation 7 implicitly assumes that the effect of the instrument's cash flows on the company's default probability is already captured by the horizon-specific cumulative default probabilities.

14. These values are based on Altman and Kishore's (1996) research, which used the postdefault secondary-market value as the measure of loss.

15. Using panel estimation methods, one can jointly estimate LGDs and lambda. By obtaining a reasonable picture of the default probability, one can then isolate the market assessment of expected recovery. This approach is a focus of my current research.

16. In practice, my colleagues and I have not found much difference in goodness of fit between the "single cash flow" and the "absorbing barrier" specifications, and because of the adjustment of the bond yields to a zero-coupon equivalent, as described later, the work in this section uses the single-cash-flow specification.

17. One problem with using such methods is that they do not account for calls arising from improvement in credit quality and thus probably understate the value of the call option for high-yield bonds. After some experimentation with alternative methods, including simply using yield to worst, however, I found that standard option adjustment provided the best available procedure.

18. An alternative procedure is to swap the bonds to a floating-rate basis and take the credit spread to be the spread over the floating base. I considered this method as an alternative but found that the two methods yielded essentially the same results.

19. Bonds with maturities greater than 10 years were eliminated because the default probabilities used here were available for only up to 5-year horizons. Using option-adjusted duration meant that 10-year bonds generally were treated as 6-7-year zero coupons, and the default probabilities were extrapolated for these cases. REIT bonds and PIKs were eliminated because their pricing appears to be anomalous relative to other bonds. Bonds with large spreads were eliminated because such bonds tend to be highly distressed or in default and their value is assumed to be driven primarily by recovery risk, not default risk.

20. The minimum, maximum, and median number of bonds on these nine different dates were, respectively, 2,098, 2,302, and 2,212.

21. The time variation in the single-parameter ratings model is also less than one might have expected intuitively because of the constancy of the agency default rates over time. The other slightly surprising result is that the rating-based model with seven parameters does somewhat worse than the rating-based model with one parameter.

22. The use of empirically based default probabilities raises the issue of in-sample versus out-of-sample results. As noted previously, the empirical default probabilities were fit in 1995 on the basis of all companies with public equity with data for 1978-1994; the results reported here involved only companies with both traded equity and debt (a much smaller and distinctive subset) and a sample period within 2001. Thus, these results have a "double" out-of-sample nature.

23. One mystery resolved by these results is that the EDF term structures begin to hump and turn down at higher quality ranges than do the spreads. In the conversion of actual default probabilities to pricing probabilities, a risk premium is added that increases with term. The combination results in shifting the transition to humped and downward-sloping term structures into a lower quality range, which is consistent with the spread data.

24. Because he carried out his work prior to the valuation studies reported here, Vasicek simply fit EDFs to spreads as a nonlinear function of term. He then used the fitted relationship to divide the sample into three groups: where the credit spread was (1) about right relative to the EDF, (2) too high, and (3) too low. The groupings would have been relatively insensitive to the exact valuation model specification; thus, their applicability to the current setting.

25. No adjustment was made for trading costs, so this differential does not reflect what could have been achieved by an actual trading strategy.

26. Misreported prices generate apparent excess returns in the following way: A misreported price makes the bond seem either under- or overpriced. A subsequent correct price (on average) will then make it appear that the misvaluation has disappeared.

27. For example, Smithson (2001) reported on a survey of banks' use of default models.

References:

Altman, Edward I. 1968. "Financial Ratios, Discriminant Analysis, and the Prediction of Corporate Bankruptcy." Journal of Finance, vol. 23, no. 4 (September):589-609.

Altman, Edward I., and Vellore M. Kishore. 1996. "Almost Everything You Wanted to Know about Recoveries on Defaulted Bonds." Financial Analysts Journal, vol. 52, no. 6 (November/December):57-64.

Altman, Edward I., R. Haldeman, and P. Narayanan. 1977. "ZETA Analysis, A New Model for Bankruptcy Classification." Journal of Banking and Finance, vol. 1, no. 1 (June):29-54.

Black, Fischer, and John Cox. 1976. "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions." Journal of Finance, vol. 31, no. 2 (May):351-367.

Black, Fischer, and Myron Scholes. 1973. "The Pricing of Options and Corporate Liabilities." Journal of Political Economy, vol. 81, no. 3 (May/June):637-659.

Bohn, Jeffrey R. 2000a. "An Empirical Assessment of a Simple Contingent-Claims Model for the Valuation of Risky Debt." Journal of Risk Finance, vol. 1, no. 4 (Spring):55-77.

_____. 2000b. "A Survey of Contingent-Claims Approaches to Risky Debt Valuation." Journal of Risk Finance, vol. 1, no. 3 (Spring):53-78.

Bohn, Jeffrey R., and Irina Korablev. 2000. "Predicting Agency Rating Upgrades and Downgrades with KMV's Expected Default Frequency (EDF)." KMV Corporation.

Carty, Lea V. 1997. "Moody's Rating Migration and Credit Quality Correlation, 1920-1996." Moody's Investors Service.

Carty, Lea V., and Jerome S. Fons. 1993. "Measuring Changes in Corporate Credit Quality." Moody's Special Report (November).

Carty, Lea V., and Dana Lieberman. 1999. "Corporate Bond Defaults and Default Rates 1938-1995." Journal of Risk Finance, vol. 1, no. 1 (Fall):87-105.

Cass, Dwight. 2000. "Moody's Launches." Risk, vol. 13, no. 6 (June):14-15.

Collin-Dufresne, Pierre, and Bruno Solnik. 2001. "On the Term Structure of Default Premia in the Swap and LIBOR Markets." Journal of Finance, vol. 56, no. 3 (June):1095-1115.

Crouhy, Michel, Dan Galai, and Robert Mark. 2001. Risk Management. New York: McGraw-Hill.

Delianedis, Gordon, and Robert Geske. 1998. "Credit Risk and Risk Neutral Default Probabilities: Information about Rating Migrations and Defaults." Working Paper #19-98, University of California at Los Angeles.

Demchak, Bill. 2000. "Modelling Credit Migration." Risk, vol. 13, no. 2 (February):99-102.

Eom, Young Ho, Jean Helwege, and Jing-Zhi Huang. 2000. "Structural Models of Corporate Bond Pricing: An Empirical Analysis." Working paper, Yonsei University.

Fama, Eugene F. 1970. "Efficient Capital Markets: A Review of Theory and Empirical Work." Journal of Finance, vol. 25, no. 2 (May):383-417.

_____. 1991. "Efficient Capital Markets: II." Journal of Finance, vol. 46, no. 5 (December):1575-1617.

Gupton, Greg M., Christopher C. Finger, and Mickey Bhatia. 1997. "CreditMetrics(TM)-Technical Document." J.P. Morgan (pp_model_20.htm).

Gurnaney, Anil. 1998. "Empirical Test of EDF, 1990-1996." Unpublished paper, KMV Corporation.

Ingersoll, Jonathon. 1977a. "A Contingent-Claims Valuation of Convertible Securities." Journal of Financial Economics, vol. 4, no. 3 (May):289-321.

_____. 1977b. "An Examination of Corporate Call Policies on Convertible Securities." Journal of Finance, vol. 32, no. 2 (May):463-478.

Jarrow, Robert A., and Donald R. van Deventer. 1999. "Practical Usage of Credit Risk Models in Loan Portfolio and Counterparty Exposure Management." In Credit Risk: Models and Management. Edited by David Shimko. London: Risk Books.

Jarrow, Robert A., D. Lando, and S. Turnbull. 1997. "A Markov Model for the Term Structure of Credit Risk Spreads." Review of Financial Studies, vol. 10, no. 2 (Summer):481-523.

Jones, P., S. Mason, and E. Rosenfeld. 1984. "Contingent Claims Analysis of Corporate Capital Structures: An Empirical Investigation." Journal of Finance, vol. 39, no. 3 (July):611-625.

Kao, Duen-Li. 2000. "Estimating and Pricing Credit Risk: An Overview." Financial Analysts Journal, vol. 56, no. 4 (July/August):50-66.

Kealhofer, Stephen. 1991. "Credit Monitor Overview." Unpublished paper, KMV Corporation.

Kealhofer, Stephen, and Matthew Kurbat. 2001. "The Default Prediction Power of the Merton Approach, Relative to Debt Ratings and Accounting Variables." Unpublished paper, KMV Corporation (Knowledge_Base/public/general/white/Merton_Prediction_Power. PDF).

Kealhofer, Stephen, Sherry Kwok, and Wenlong Weng. 1998. "Uses and Abuses of Bond Default Rates." CreditMetrics Monitor (First Quarter):37-55.

Lim, Francis. 1999. "Comparative Default Predictive Power of EDFs and Agency Debt Ratings." Unpublished paper, KMV Corporation (December).

Merton, Robert C. 1973. "Theory of Rational Option Pricing." Bell Journal of Economics, vol. 4, no. 1 (Spring):141-183.

_____. 1974. "On the Pricing of Corporate Debt: The Risk Structure of Interest Rates." Journal of Finance, vol. 29, no. 2 (May):449-470.

Miller, Ross. 1998. "Refining Ratings." Risk, vol. 11, no. 8 (August):97-99.

Ogden, Joseph P. 1987. "Determinants of the Ratings and Yields on Corporate Bonds: Tests of the Contingent Claims Model." Journal of Financial Research, vol. 10, no. 4 (Winter):329-339.

Pitts, C.G., and M.J. Selby. 1983. "The Pricing of Corporate Debt: A Further Note." Journal of Finance, vol. 38, no. 4 (September):1311-13.

Riley, Lance. 1992. "Predicted versus Actual Default Rates." KMV Corporation.

Sarig, Oded, and Arthur Warga. 1989. "Some Empirical Estimates of the Risk Structure of Interest Rates." Journal of Finance, vol. 44, no. 4 (December):1351-60.

Smithson, Charles W. 2001. "Managing Loans: The State of Play." Credit (December/January):26-31.

Standard & Poor's. 1991. "S&P's Corporate Finance Criteria."

Vasicek, Oldrich A. 1984. "Credit Valuation." Unpublished paper, KMV Corporation.

_____. 1995. "EDFs and Corporate Bond Pricing." Unpublished paper, KMV Corporation.

_____. 1997. "Credit Valuation." NetExposure, vol. 1 (October); reprinted in 1998 on CD-ROM in Derivatives: Theory and Practice of Financial Engineering. Edited by P. Wilmott. London: J. Wiley & Sons.

_____. 2001. "Actual and Risk-Neutral Probabilities of Default." Unpublished paper, KMV Corporation.

Wei, David Guoming, and Dajiang Guo. 1997. "Pricing Risky Debt: An Empirical Comparison of the Longstaff and Schwartz and Merton Models." Journal of Fixed Income, vol. 7, no. 2 (September):8-28.

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