Executive Chapter Summaries

CHAPTER 1 ESTIMATING DEFAULT PROBABILITIES IMPLICIT IN EQUITY PRICES

Tibor Janosi, Robert Jarrow, and Yildiray Yildirim

This chapter uses a reduced-form credit risk model to estimate default probabilities implicit in equity prices. The equity return model developed includes the possibility of default, market risk premiums, and price bubbles. For a cross section of firms, this equity return model is estimated using monthly returns in a time-series regression. Three conclusions are obtained. First, the analysis supports the feasibility of estimating default probabilities implicit in equity returns. This new estimation procedure provides a third alternative to using either structural models with equity prices or reduced-form credit risk models with debt prices for estimating default probabilities.

Second, we find that equity returns contain a bubble component not captured by the traditional Fama–French four-factor model for equity’s risk premium. This bubble component, proxied by the firm’s price-earnings ratio, is significant for many of the firms in our sample.

Third, due to noise introduced in equity returns by price bubbles and market risk premium, the estimated default probabilities may confound these quantities, giving biased estimates with large standard errors.

Indeed, the default probabilities obtained herein are larger than those previously obtained using either logit models based on historical data or those obtained implicitly from debt prices. By extrapolation, these results cast additional doubt on the precision of default probabilities obtained using structural models with equity prices.

CHAPTER 2 PREDICTIONS OF DEFAULT PROBABILITIES IN STRUCTURAL MODELS OF DEBT

Hayne E. Leland

This chapter examines default probabilities predicted by alternative “structural” models of risky coporate debt. We focus on default probabilities rather than credit spreads because (1) they are not affected by additional market factors such as liquidity and tax differences, and (2) prediction of the relative likelihood of default is often stated as the objective of bond ratings. We have three objectives:

1. To distinguish “exogenous default” from “endogenous default” models
2. To compare these models’ predictions of default probabilities, given common inputs.
3. To examine how well these models capture actual average default frequencies, as reflected in Moody’s Investors Service (2001) corporate bond default data 1970–2000.

We find the endogenous and exogenous default boundary models fit observed default Frequencies very well for horizons 5 years and longer, for both investment-grade and non-investment-grade ratings. Shorter-term default frequencies tend to be underestimated. This suggests that a jump component should be included in asset value dynamics. Both types of structural models fit available default data equally well. But the models make different predictions about how default probabilities and recovery rates change with changes in debt maturity or asset volatility. Further data and testing will be needed to test these differences. Finally, we compare and contrast these structural models’ default predictions with a simplified version of the widely used Moody’s–KMV “distance to default” model described in Crosbie and Bohn (2002).

CHAPTER 3 SURVEY OF THE RECENT LITERATURE: RECOVERY RISK

Sanjiv R. Das

This chapter surveys a selection of recent working papers on recovery rates, providing a framework for extant research. Simpler versions of models are also presented with a view to aid accessibility and pedagogical presentation. Despite the obvious empirical difficulties encountered with recovery rate data, modeling advances are making possible better quantification and measurement of recovery, and will result in innovative contracts to span this risk.

CHAPTER 4 NON-PARAMETRIC ANALYSIS OF RATING TRANSITION AND DEFAULT DATA

Peter Fledelius, David Lando, and Jens Perch Nielsen

Is the current rating of a bond issuer sufficient to determine the probability of default or the probability of a rating change? The empirical evidence says no. Previous history matters, and probabilities depend on whether the firm entered into its current rating class through a downgrade or an upgrade, and on the amount of time spent in the current rating class.

Non-parametric analysis of rating transition intensities is a powerful way of visualizing such effects and is therefore useful both for quickly understanding the behavior of a rating system and for exploring data before setting up a full statistical model. In this chapter we illustrate the use of non-parametric and smoothing methods for analyzing rating transitions by showing how the time spent in a particular rating class and the direction of the move into this class influence transition intensities away from the class.

CHAPTER 5 VALUING HIGH-YIELD BONDS: A BUSINESS MODELING APPROACH

Thomas S. Y. Ho and Sang Bin Lee

This chapter provides a model for valuing high-yield bonds. The model asserts that a corporate bond is a contingent claim on the firm, which in turn is a contingent claim on the business risks. Therefore, the bond model takes the debt structure, capital structure, operating leverage, and other aspects of a business model into account. The model is empirically testable and the chapter shows that the model can better explain some empirical results.

A corporate bond valuation model that takes a business modeling approach has many applications. In corporate finance, we can identify the impact of the operating leverage and the financial leverage on the risk transfer to the stakeholders of the firm. The model can be used for the design of the debt structure in relation to the business model of the firm. In equity research, the model solves for the appropriate cost of equity of the business risk. As a result, the model enables us to compare the stock value of firms in a similar business but with different capital structure, debt structure, and operating leverage. In bond research, we can value the senior and junior corporate bonds within the debt package, taking the business model into account. The model can be used for investment in stocks or bonds and in corporate finance in capital structure decisions.

CHAPTER 6 STRUCTURAL VERSUS REDUCED-FORM MODELS: A NEW INFORMATION-BASED PERSPECTIVE

Robert A. Jarrow and Philip Protter

For modeling credit risk, two classes of models exist: structural and reduced-form. Structural models often assume that default occurs when the firm’s asset value hits a barrier, whereas reduced-form models use a hazard rate framework to model default. These models are viewed as disjoint and competing, and there is heated debate as to which class of models is best for predicting default and/or pricing credit-risky instruments.

This chapter shows that structural and reduced-form models are not disjoint model types, but rather the same model containing different informational assumptions. Structural models assume knowledge held by the firm’s managers: continuous observations of the firm’s asset values and liabilities. In contrast, reduced-form models assume the knowledge observed by the market: the information generated by a set of state variables, and the firm’s default time and recovery rate process. It is shown that structural models can be transformed into reduced-form models as the information becomes less refined—from that observed by the firm’s management, to that which is observed by the market.

Given this insight, the current debate in the credit risk literature about these two model types needs to be redirected. The debate should be focused on whether or not the model should be based on the information set observed by the market. For pricing and hedging credit risk, the information set observed by the market is the relevant one. This is the information set used in the economy to determine prices. For internal risk management purposes with respect to its own credit risk, however, a structural model may be preferred.

CHAPTER 7 REDUCED FORM VERSUS STRUCTURAL MODELS OF CREDIT RISK: A CASE STUDY OF THREE MODELS

Navneet Arora, Jeffrey R. Bohn, and Fanlin Zhu

In this chapter we compare two structural models [basic Merton and Vasicek–Kealhofer (VK) as implemented by Moody’s–KMV] and one reduced-form model [Hull–White (HW)] of credit risk. In order to make the model testing relevant to practical implementation of these models, our evaluation criteria must address concerns faced by practitioners in practice (as opposed to theoretical concerns.) We propose that two useful purposes for credit models are default discrimination and relative value analysis. Default discrimination is relevant for quantitative risk and portfolio management. Relative value analysis is relevant for mark-to-market exercises for portfolio management and decision support in a credit trading environment. We test the ability of the Merton and VK models to discriminate defaulters from nondefaulters based on default probabilities generated from information in the equity market. We test the ability of the HW model to discriminate defaulters from nondefaulters based on default probabilities generated from information in the bond market. We find the VK and the HW models exhibit comparable accuracy ratios and substantially outperform the simple Merton model. We also test the ability of each model to predict spreads in the credit default swap (CDS) market as an indication of each model’s strength as a relative value analysis tool. We find the VK model tends to do the best across the full sample and relative subsamples except for cases where an issuer has many bonds in the market. In this case, the HW model tends to do the best. The empirical evidence will assist market participants in determining which model is most useful based on their “purpose in hand.” Note that these tests end up producing evaluations of not only the modeling approaches, but also the different data sources. Models relying on poor or scarce data are not that useful regardless of how good they seem in theory.

On the structural side, a basic Merton model is not good enough; appropriate modifications to the framework make a difference. On the reduced-form side, the quality and quantity of data make a difference; many traded issuers will not be well modeled in this way unless they issue more traded debt. In addition, bond spreads at shorter tenors (less than two years) tend to be less correlated with CDS spreads. This makes accurate calibration of the term structure of credit risk difficult when relying on bond data. The widespread availability and reliability of equity data tend to produce better results for the more sophisticated structural model evaluated in these tests. In cases where an issuer has many outstanding bonds and data for these bonds are widely available, the reduced-form model produces better results. For practitioners looking across the broad cross section of traded credit instruments, the data requirements for robust reduced-form modeling and the availability of robust equity-based measures should inform discussions on which modeling approach to use. Moreover, users of reduced-form models looking to price other credit-risky securities such as CDSs should bear in mind the potential impact on bond spreads of other risks resulting from interest rate movements and changes in liquidity. These effects can differ across size and spread levels, thereby distorting the performance of these models. A model such as VK is relatively more stable in its performance across various categories by size and spreads. This model’s strength partially owes to its structural framework that uses equity data—which are less contaminated by other risks—and partially owes to its more sophisticated implementation. The overall results emphasize the importance of empirical evaluation when assessing the strengths and weaknesses of different types of credit risk models.

CHAPTER 8 IMPLICATIONS OF CORRELATED DEFAULT FOR PORTFOLIO ALLOCATION TO CORPORATE BONDS

Mark B. Wise and Vineer Bhansali

The possibility of correlated defaults makes the problem of optimal allocation to corporate bonds interesting for both plain vanilla corporate bond investors and investors in structured credit products. Because the loss distribution of these portfolios does not have to be normal, the applicability of classic mean–variance allocation machinery is called into question. In this chapter we show that under very general assumptions for the utility function of investors and a large range of parameters, the optimal allocation decision of an investor can still be well approximated by the first three moments of the loss distribution. The intuition developed by our numerical results is valuable for investors who are looking to invest in corporate bonds or structured products such as collateralized debt obligations (CDOs) or their tranches.

This chapter provides a road map for setting these fees. It draws on a marketing research technique, conjoint analysis, and straightforward optimization procedure to guide fund managers toward making better decisions. These decisions have potential benefits for both mutual fund companies in terms of increased profitability and for investors in terms of increased utility.

To demonstrate our approach we conducted a conjoint analysis with 50 mutual fund investors. We find that, although most mutual fund investors strongly prefer low front-end loads, significant opportunities for price discrimination exist in this marketplace. Our model uncovers the potential for multiple efficient fee structures for different classes of fund shares, and links this potential to observable customer characteristics. Investment management companies can use these observed linkages to better target their pricing and overall marketing efforts.

CHAPTER 9 CORRELATED DEFAULT PROCESSES: A CRITERION-BASED COPULA APPROACH

Sanjiv R. Das and Gary Geng

We are witnessing an explosive growth in structured products based on baskets comprising a large number of defaultable bonds. These take the form of collateralized debt obligations (CDOs) or basket default swaps, and many others. Modeling a large system of correlated default in order to price and manage the risk of these new products is thus a fast-growing area of research.

The joint movement of default probabilities is not usually governed by a multivariate normal distribution, which is often assumed for tractability. Moving beyond the normal distribution requires more sophisticated modeling. Copulas are a general and facile technical device for the modeling of correlated default processes with varying assumptions on the joint distribution. Although there is a growing literature on the technical modeling of default using copulas, the practicalities involved in fitting copulas to a default system have not received empirical attention. This chapter seeks to fill this gap. We develop a methodology to calibrate the joint default process of hundreds of issuers. To determine the appropriate choice of the joint default process, we propose a new metric. This metric accounts for three different aspects of default correlation: (1) level, (2) asymmetry, and (3) tail-dependence and extreme behavior.

Given a choice of copula, the joint distribution of default probabilities may be estimated, and the system calibrated for simulation purposes. This chapter goes a step further and provides a methodology for comparison among copulas as well. The approach is based on a new metric that is practically motivated, and may be used by practitioners in deciding which copula to employ in their models.

The designers of CDOs will be able to improve their pricing of CDO tranches. Rating agencies will be able to sharpen their assessments of securities based on correlated default. Traders in CDOs who are engaged in capital arbitrage will have a range of models to choose from. Insurance companies and depository institutions will be able to fit copulas to both the asset and liability sides of their balance sheets, and assess the joint default risk of the firm in entirety. In addition, regulators may use copulas to model systemic risk by aggregating correctly across entities. This approach thus has practical value for many market participants.