Quantifying Credit Risk I: Default Prediction

Quantifying credit risk I: Default prediction

Stephen Kealhofer
8,184 words
1 January 2003
Financial Analysts Journal
30
Volume 59, Issue 1; ISSN: 0015-198X
English
Copyright (c) 2003 ProQuest Information and Learning. All rights reserved. Copyright Association for Investment Management and Research Jan/Feb 2003

Until the 1990s, corporate credit analysis was viewed as an art rather than a science because analysts lacked a way to adequately quantify absolute levels of default risk. In the past decade, however, a revolution in credit-risk measurement has taken place. The evidence from this research presents a compelling case that the conceptual approach pioneered by Fischer Black, Robert Merton, and Myron Scholes provides a powerful practical basis for measuring credit risk.

One of the major frontiers 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 to implement the Merton model (notably, Jones, Mason, and Rosenfeld 1984 and Ogden 1987) produced discouraging results, and even today, researchers continue to reproduce the original negative results (e.g., Jarrow and Van Deventer 1999). In 1984, however, Vasicek took a novel approach to implementation of the Merton model that has proven to have considerable success in measuring credit risk. This version of the Merton model has been extended by KMV Corporation to become a de facto standard for default-risk measurement in the world of credit risk.1

This first of two companion articles on the empirical research on the KMV model in the past decade explains the differences in implementation between the KMV and Merton models and reviews studies comparing the performance of the KMV model against agency debt ratings in predicting default.2 Part II will present results on the use of the model to value corporate debt relative to alternative approaches.

The thesis of the articles is that the BlackScholes-Merton approach, appropriately executed, represents 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 prices, it produces empirical estimates of default probability that outperform well-accepted public benchmarks, such as agency debt ratings, in predicting default. Finally, in contrast to earlier findings, this study finds that the Black-Scholes-- Merton approach provides a better empirical fit to the value of corporate bonds than more conventional approaches that use agency bond ratings.

The KMV and Merton Models

The approach Black and Scholes developed can be illustrated in a simplified case. Suppose a company has a single asset consisting of 1 million shares of Microsoft stock. Furthermore, it has a single fixed liability-a one-year discount note with a par amount of $100 million-and is otherwise funded by equity. In one year's time, the market value of the company's business will either be sufficient to pay off the note or it will not, in which case the company will default. One can observe that the equity of the company is logically equivalent to 1 million call options on Microsoft stock, each with an exercise price of $100 and a maturity of one year.

The implication of this illustration is that the equity of a company is like a call option on the company's underlying assets. The value of the equity thus depends on, among other things, the market value of the company's assets, their volatility, and the payment terms of the liabilities. Implicit in the value of the option is a measure of the probability of the option being exercised; for equity, it is the probability of not defaulting on the company's liability.3

Figure 1 provides an illustration of these concepts. The horizontal axis represents time, beginning with the current period ("Today") and looking into the future. The vertical axis depicts the market value of the company's assets. As of the current period, the assets have a single, determinable value, as shown on the vertical axis, but one year from now, a range of asset values is possible, and their frequency distribution (shown in Figure 1 on its side) gives the likelihood of various asset values one year in the future. The most likely outcomes are nearest to the starting value, with much larger or smaller values less likely. The mean is shown by the dashed line. The likelihood of extreme outcomes depends on the volatility of the assets-the more volatile, the greater the probability of extreme outcomes. The dotted horizontal line shows the par amount of the liability due in one year. If the company's asset value in one year is less than the amount of the liability, the company will default. Note that this decision is an economic decision: The equity owners could put additional money into the company, but that decision would be irrational because the money would go to pay creditors; if the owners defaulted, they would not be required to put in additional money and they could use this money for their own benefit rather than giving it to the creditors. The probability of default is thus given by the area under the frequency distribution below the default point, which represents the likelihood of the market value of the company's assets in one year being less than what the company owes. It is immediately obvious that the probability of default will increase if the company's market value of assets today decreases, if the amount of liabilities increases, or if the volatility of the assets' market value increases. These three variables are the main determinants of the company's default probability. Figure 1.

What Black, Scholes, and Merton actually proposed is a general framework for valuing contingent claims. There is no single Merton model; indeed, the KMV model is largely a variant of Merton models. The KMV model differs from the standard variants in some significant ways, however, and it also includes a scheme for empirical default-probability measurement that lies outside the standard approach. To appreciate these differences, one must begin with the canonical Merton model.

The classic exposition of the Merton model is Merton (1974). This model has the following characteristics:

* The company has a single debt liability, has equity, and has no other obligations.

* The liability promises a continuous fixed coupon flow and has an infinite maturity.

* The company makes no other cash payouts (e.g., equity dividends).

Under the assumption that the market value of the company's assets evolves as a lognormal process, Merton showed that this model can be solved for a closed-form solution for the value of the company's debt. The aim of the model, and much subsequent work based on this model, is to obtain a valuation equation for the company's debt.

The KMV model, building on previous work by Black and Cox and by Ingersoll (1977a) begins with a somewhat more general characterization of the company's capital structure. The most important distinction between the models, however, is the KMV model's primary focus on the probability of default of the company as a whole, rather than valuation of the debt. The KMV model has the following characteristics:4

* The company may have, in addition to common equity and possibly preferred stock, any number of debt and nondebt fixed liabilities.

* The company may have warrants, convertible debt, and/or convertible preferred stock.

* Obligations may be short term, in which case they are treated as demandable by creditors, or long term, in which case they are treated as perpetuities.

* Any and all classes of liability, including equity, may make fixed cash payouts.

* If the market value of the company's assets falls below a certain value (the default point), the company will default on its obligations; this default point depends on the nature and extent of the company's fixed obligations.

* Default is a company-wide event, not an obligation-specific event.

Whereas the objective in the Merton model is the valuation of the company's debt based on the company's asset value and volatility, the focus in the KMV model is on the relationship between the company's equity characteristics and its asset characteristics. Given the asset characteristics (i.e., value and volatility) and given the company's default point, one can use the KMV model to immediately calculate a simple, robust measure of the company's default risk-the number of standard deviation moves required to bring the company to the default point within a specified time horizon.5

Distance to default, DD(h), or the number of standard deviations to the default point by horizon h, is an ordinal measure of the company's default risk. As such, it provides a simple and robust measure of default risk. Mathematically, it is calculated as

From a certain standpoint, these model differences are not significant. In terms of practical implementation, however, they are critical. Researchers using the Merton model to investigate the pricing of corporate debt obtained poor results. Research based on the distance to default as the measure of default risk has yielded excellent results.

Why focus on default-risk measurement rather than debt valuation? The answer is that debt valuation should implicitly contain default-risk measurement. Thus, if the default-risk measurement is verified as correct, the subsequent debt valuation should also be correct. Because earlier efforts at valuation had failed, a focus on default-risk measurement made sense because it could be independently tested. Any insights garnered from that testing could then be used to diagnose and, potentially, resolve the valuation problems.

The KMV model focus on default-risk measurement leads to another important aspect of the model. As noted previously, the distance-to-- default measure is an ordinal measure. For valuation purposes, however, one needs an absolute measure, an explicit probability of default. To date, other researchers have obtained default probabilities by using the assumed lognormal asset-value distribution of the Merton approach.

The assumption of log-normality cannot be evaluated without determining the actual default experience of a large, well-defined population of companies. KMV has been tracking the default experience of all publicly traded companies in the United States since 1973. These data made possible a comparison of the default probabilities calculated from the lognormal distribution with actual realized default rates. This comparison established that small but important differences exist between the theoretical and actual default rates. For instance, under the normality assumption, any company more than about four standard deviations from its default point would have essentially zero probability of default. In actuality, the default probability for such companies is meaningfully higher, about 0.5 percent. These small numeric deviations in tail probabilities between the normal distribution and the empirical distribution translate into economically significant differences in terms of default risk. For instance, a company that is four standard deviations from the default point as measured by the normal density would be better than AAA in quality, whereas with an actual 0.5 percent default probability, it would not even be investment grade.8

With the use of the KMV default database, we found that we could measure the empirical distribution with sufficient accuracy that the empirical probabilities could be substituted for the theoretical probabilities. This measurement relies on the distance to default as a "sufficient statistic" for the default risk, so all the default data for companies with similar DDs can be pooled. Put differently, the differences between individual companies are expected to be reflected in their asset values, their volatilities, and their capital structures, all of which are accounted for in their DDs. The estimation need not be performed on separate subsamples-for instance, by industry or size. Where feasible, results for subsamples have been compared with the overall pooled results, and no statistically significant differences have been observed.9

The result of this process is the KMV EDF(TM) (expected default frequency) credit measure. The EDF is the probability of default within a given time period. It is a monotone function of the distance to default, so it preserves the ordinal properties of the DD measure while providing a cardinal measure of default probability.

KMV Model Default-Predictive Power

Two types of tests on the predictive power of the Merton approach (as implemented in the KMV model) have been carried out. The first type, "power tests," characterizes the relative ability of a default-risk measure to correctly identify companies that subsequently default versus incorrectly identifying companies as being likely defaulters that do not default. The second type of test, "intracohort analysis," provides a method for evaluating whether the differences between two measures are the result of additional information or are simply noise.

Power Tests. Default is a binary event; it either occurs or it does not. Most measures of default risk-for instance, agency debt ratings-- are indexes that indicate relative likelihood of default. One can translate ordinal measures, such as indexes, into binary choices for purposes of testing by interpreting them as an acceptance/ rejection criterion. To do so, one determines a cutoff value, v, and accepts as "food" all companies with an index value above v.10

Given a cutoff value v, two types of errors can be made. A Type I error is identifying a company as good (because it has a value above the cutoff) but it subsequently defaults. A Type II error is identifying a company as bad (because it has a value at or below the cutoff) but it subsequently does not default.

Three aspects of testing default-risk measures deserve special attention. They concern arbitrary metrics, error trade-off, and sample dependence.

* Arbitrary metrics. Almost by definition, different default-risk measures are based on different metrics. Thus, cutoff values are unrelated from one metric to another. Comparing different indexes at arbitrary cutoff values is generally meaningless.

* Error trade-off. The levels of Type I and Type II errors are related to each other and to the level of the cutoff. Using a very high cutoff minimizes Type I error, tl, but maximizes Type II error, t2. Thus,

t1(v) is decreasing in v;

t2(v) is increasing in v.

* Sample dependence. The levels of error depend on the particular sample tested. In general, one cannot compare errors from one sample with errors from another sample. For instance, one sample may consist of the larger companies or the companies with longer histories. Differences of this type almost invariably translate into different average default rates and different levels of error.

These three observations have a number of implications. For instance, because of error tradeoff and sample dependence, stating the amount of Type I error in isolation, as in "our internal risk ratings are 99 percent accurate," is meaningless. If the underlying sample had only a 1 percent default rate, then a measure that passed the entire sample as good would be 99 percent accurate and also have zero Type II errors!

A less obvious but equally fallacious result occurs when default-risk measures are tested on different samples but the results are reported as comparable. For instance, suppose an analyst tests two models on the same sample of companies; one model calls for three years of financial data, and the other, for only the current financial data. Because some companies will drop out of the database in a three-year period, fewer companies will have three years of financial data. The data for these survivors will be somewhat less volatile, on average, and the sample will have a lower average default rate. Rather than testing each model on the subset of companies with data available for that particular model, one must test the two models on exactly the same set of companies-in this case, the subset of companies with both historical and current data available.

The problem of arbitrary metrics can be overcome by comparing the Type I error rates of the models for which the cutoffs have been set to produce the same levels of Type II error. (Equivalently, one could compare Type II errors holding Type I errors constant.) Because, in practice, one is interested only in errors, not arbitrary cutoff values, the relevant issue is how much error of one type exists for a given amount of the other type of error.

Mathematically, the "power curve" for index i, p^sub i^(x), is graphed as

The purpose of the power curve is to fully characterize both types of errors that a defaultprediction model makes-namely, failing to predict a company that does default and predicting a default that does not occur. One economic interpretation of the power curve can be understood by thinking about the cutoff value as representing the value a company must have to be approved for a loan. A uniformly more powerful measure is the one that results in a lower default rate for approved loans for any given refusal rate.