The Credit Market Handbook

In this paper, we study the empirical performance of structural credit risk models by examining the default probabilities calculated from these models with different time horizons. The parameters of the models are estimated from firm's bond and equity prices. The models studied include Merton (1974), Merton model with stochastic interest rate, Longstaff and Schwartz (1995), Leland and Toft (1996) and Collin-Dufresne and Goldstein (2001). The sample firms chosen are those that have only one bond outstanding when bond prices are observed. We first find that when the Maximum Likelihood estimation, introduced in Duan (1994), is used to estimate the Merton model from bond prices, the estimated volatility is unreasonable high and the estimation process does not converge for most of the firms in our sample. This shows that the Merton (1974) is not able to generate high yields to match the empirical observations. On the other hand, when equity prices are used as input we find that the default probabilities predicted for investment-grade firms by Merton (1974) are all close to zero. When stochastic interest rates are assumed in Merton model, the model performance is improved. The models of Longstaff and Schwartz (1995) with constant interest rate as well as the Leland and Toft (1996) provide quite reasonable predictions on real default probabilities when compared with those reported by Moody's and S&P. However, Collin-Dufresnce and Goldstein (2001) predict unreasonably high default probabilities for longer time horizons.

Introduction Default is triggered by a firm's failure to meet its financial obligations. Default probabilities and changes in expected default frequencies affect markets participants, such as investors and lenders, since they assume responsibility for the credit risk of their investments. The lack of a solid economic understanding of the factors that determine bankruptcy makes explanation and prediction of default difficult to assess. However, the accuracy of these predictors is essential for sound risk management and for evaluation of the vulnerability of corporations and institutional lenders. In recognition of this, the new capital adequacy framework (Basel II) envisages a more active role for banks in measuring the default risk of their loan books. The need for reliable measures of default or credit risk is clear to all. The accounting and finance literature has produced a variety of models attempting to predict or measure default risk. There are two primary types of models that describe default processes in the credit risk literature: structural models and reduced-form models. Structural models use the evolution of a firm's structural variables, such as asset and debt values, to determine the timing of default. Merton's model (1974) is considered the first structural model. In Merton's model, a firm defaults if, at the time of servicing the debt at debt maturity, its assets are below its outstanding debt. A second approach within the structural framework was introduced by Black and Cox (1976). In this approach default occurs when a firm's asset value falls below a certain threshold.

In this work, we introduce a general framework for incorporating stochastic recovery into structural models. The framework extends the approach to recovery modeling developed in Cohen and Costanzino (2015, 2017) and provides for a systematic way to include different recovery processes into a structural credit model. The key observation is a connection between the partial information gap between firm manager and the market that is captured via a distortion of the probability of default. This last feature is computed by what is essentially a Girsanov transformation and reflects untangling of the recovery process from the default probability. Our framework can be thought of as an extension of Ishizaka and Takaoka (2003) and, in the same spirit of their work, we provide several examples of the framework including bounded recovery and a jump-to-zero model. One of the nice features of our framework is that, given prices from any one-factor structural model, we provide a systematic way to compute corresponding prices with stochastic recovery. The framework also provides a way to analyze correlation between Probability of Default (PD) and Loss Given Default (LGD), and term structure of recovery rates.

The main idea of this paper is to examine the dependence between the probability of default (PD) and the recovery rate (RR). For the empirical methodology, we use the bootstrapped quantile regression and the simultaneous quantile regression for a sample of 17 Greek banks listed in Athens Exchange over the period of study from January 02, 2006 to December 31, 2012. The measurement of this dependence is determined by using seven indicators such as the probability of default, the recovery rate, the number of defaults, the expected value of losses, the growth rate of GDP in Greece, and three dummy variables (the exit of another firm of the Athens Exchange, the new firm is listed in the Athens Exchange, and the date of the failure of Greece). The main empirical results show that the probability of default and the recovery rate are inversely related. Based on this result, the banks are obliged to maximize their recovery rate to reduce their probability of default.

In this paper, we investigate whether the theoretical default probability measures calculated from Merton's (1974) structural credit risk model can provide a better way to explain and predict credit rating than traditional statistical models. The empirical results suggest that Merton's theoretical default measure is not a sufficient statistic of equity market information concerning credit quality. By including the market value of the firm as an independent variable we can improve both in-sample fitting and out-of-sample predictability of credit ratings. Moreover, the empirical performance of this hybrid model is very similar to those simple statistical model. As a result, we conclude that structural models hardly provide any additional capability in capturing credit risk. Our empirical results show that instead of using the firm value only through the debt leverage ratio, as suggested in the structure models, one should include the market value of the firm as a separate factor affecting default probability when building credit risk models.

Credit risk is related to the ability of a corporation or an individual to honor a financial obligation. A typical example is the default risk embedded in corporate bond markets. Structural models of credit risk are specifically designed to describe corporate credit risk using the capital structure of a firm. The capital structure comprises the firm's underlying assets, equities, and debts. A comprehensive approach even takes into account the optimal allocation of the firm's assets, the dividend policy of the management, and protective covenants of bond holders. The Moody's KMV, a corporation owned by the Moody's Investor Services, is the first rating agency to put a structural model into commercial use since the early 1990s. Popular structural models include, but are not limited to, the Merton model, the Black–Cox model, the Leland–Toft model, and the KMV approach. In this article, a brief overview of the formulation of a structural approach, specifications within the aforementioned models, and the implementation issues is given.

This paper presents a robust test of Merton's structural model for credit risk that does not depend on either estimated parameters for the firm's value or estimated default probabilities. We derive a test for the consistency of the changes in observed debt and equity prices (positive or negative changes) with the Merton model. For all firms selected and for all debt issues examined, the evidence strongly rejects Merton's structural model. employed to manage credit risk in banks and bond portfolios. Merton's model invokes the arbitrage free pricing methodology in friction- less and competitive markets. The key characteristic of Merton's model is that the underlying state variable that determines a firm's default is the value of its assets. Default free interest rates are assumed to be constant. Given a specific balance sheet for the firm (a fixed and static structure), at the maturity date of its short-term debt (a discount bond), if the firm's value falls below the face value of the short-term debt, then default occurs. In the event of default, the payoffs to the firm's liabilities follow absolute priority (written into the debt's covenants). Under this structure, the firm's equity is analogous to a call option on the firm's assets. As is well known, Merton's model has four empirical difficulties that make its implementation problematic:

The chapter is focused on the structural models of credit risk introducing basic concepts of risk-neutral world, as well as models and different options for the credit risk quantification. An important part is also the introduction of structural approach for credit risk modeling. Furthermore, the chapter presents basic division of structural models and then presents mathematical derivation of individual apparatuses of models. Among tested models are Merton model, KMV model, Black-Cox model, and Credit Grades model. The practical part is focused on the application of these models under the conditions of local emerging market—Slovakia. Additionally, it pointed out the connection between default probability and credit spreads generated with the use of default mode credit risk models. The main objective is to adjust credit risk model to real market data.

Credit risk is related to the ability of a corporation or an individual to honor a financial obligation. A typical example is the default risk embedded in corporate bond markets. Structural models of credit risk are specifically designed to describe corporate credit risk using the capital structure of a firm. The capital structure comprises the firm's underlying assets, equities, and debts. A comprehensive approach even takes into account the optimal allocation of the firm's assets, the dividend policy of the management, and protective covenants of bond holders. The Moody's KMV, a corporation owned by the Moody's Investor Services, is the first rating agency to put a structural model into commercial use since the early 1990s. Popular structural models include, but are not limited to, the Merton model, the Black–Cox model, the Leland–Toft model, and the KMV approach. In this article, a brief overview of the formulation of a structural approach, specifications within the aforementioned models, and the implementation issues is given.

There is strong evidence that structural models of credit risk significantly underestimate both credit yield spreads and the probability of default if the value of corporate assets follows a diffusion process. Adding a jump component to the firm value process is a potential remedy for the underestimation. However, there are very few empirical studies of jump-diffusion (or Levy) structural models in the literature. The major challenge is the estimation of hidden variables, such as the firm value, volatility, and parameters of the jump component, as the value of corporate assets is not directly observable. In practice, parameters and the value of the firm should be estimated using the market values of equities. This paper provides a promising estimation method for jump-diffusion processes in structural models that are based on observed stock data. We show that the traditional estimation methods for structural models, the variance-restriction method and maximum likelihood estimation, fail when jumps appear in credit risk models. We then propose a penalized likelihood approach and devise a corresponding expectationmaximum algorithm. The approach is applied to the jump-diffusion processes of Merton (1976) and Kou (2002) and the performance is examined through a series of simulations and empirical data.

The classic credit risk structured model assumes that risky asset values obey geometric Brownian motion. In reality, however, risky asset values are often not a continuous and symmetrical process, but rather they appear to jump and have asymmetric characteristics, such as higher peaks and fat tails. On the other hand, there are real Knight uncertainty risks in financial markets that cannot be measured by a single probability measure. This work examined a structural credit risk model in the Lévy market under Knight uncertainty. Using the Lévy–Laplace exponent, we established dynamic pricing models and obtained intervals of prices for default probability, stock values, and bond values of enterprise, respectively. In particular, we also proved the explicit solutions for the three value processes above when the jump process is assumed to follow a log-normal distribution. Finally, the important impacts of Knightian uncertainty on the pricing of default probability and stock values of enterprise were studied through numerical analysis. The results showed that the default probability of enterprise, the stock values, and bond values were no longer a certain value, but an interval under Knightian uncertainty. In addition, the interval changed continuously with the increase in Knightian uncertainty. This result better reflected the impact of different market sentiments on the equilibrium value of assets, and expanded decision-making flexibility for investors.

In this paper, we consider the default probabilities caused by a jump or by oscillation under a structural credit risk model with jumps. We study the Laplace transforms of the times of default caused by a jump and by oscillation. We derive integro-differential equations and obtain some closed-form expressions for them. By inverting them, we numerically investigate the contributions of the jump component and the diffusion component to the default under a certain choice of the jump size distribution.

This paper derives a closed-form expression for the default probability and the default correlation of firms under a structural model of credit risk. Specifically, the underlying firms are assumed to have the value process driven by a Hawkes jump-diffusion model with the continuous parts of the trajectories being driven by correlated Brownian motions, while the jumps being driven by Hawkes processes having general structure of the exciting functions. The proposed framework takes into account the numerically observed facts about the default, i.e., clustering and unexpectedness. Furthermore, the default barriers are assumed to be stochastic in nature and modeled as stochastic processes, affected by common factors reflecting the systematic risk. A sensitivity analysis of default probability and correlation is conducted to investigate the impact of jump risk, clustering, and stochastic default barriers. These numerical studies demonstrate that jump clustering increases the default probability but reduces the correlation of defaults.

OF THE THESIS Structural Credit Risk Models in Banking with Applications to the Financial Crisis by Michael B. Imerman Thesis directors: Professors Ren-Raw Chen and Ben J. Sopranzetti This dissertation uses structural credit risk models to analyze banking institutions during the recent financial crisis. The first essay proposes a dynamic approach to estimating bank capital requirements with a structural credit risk model. The model highlights the fact that static measures of capital adequacy, such as a fixed percentage of assets, do not capture the true condition of a financial institution. Rather, a dynamic approach that is forward-looking and tied to market conditions is more appropriate in setting capital requirements. Furthermore, the structural credit risk model demonstrates how market information and the liability structure can be used to compute default probabilities, which can then be used as a point of reference for setting capital requirements. The results indicate that capital requirements may be timevarying and conditional upon the health of the particular bank and the financial system as

This paper derives unbiased capital allocation rules for portfolios in which credit risk is driven by a single common factor and idiosyncratic risk is fully diversified. The methodology for setting unbiased capital allocations is developed in the context of the Black-ScholesMerton (BSM) equilibrium model. The methodology is extended to develop an unbiased capital allocation rule for the Gaussian ASRF structural model of credit risk. Unbiased capital allocations are shown to depend on yield to maturity as well as probability of default, loss given default, and asset correlations. Unbiased capital allocations are compared to capital allocations that are set equal to unexpected loss in a Gaussian credit loss model—an approach that is widely applied in the banking industry and used to set minimum bank regulatory capital standards under the Basel II Internal Ratings Based (IRB) approach. The analysis demonstrates that the Gaussian unexpected loss approach substantially undercapitalizes portfolio credit risk relative to an unbiased capital allocation rule. The results include a suggested correction for the IRB capital assignment function. The corrected capital rule calls for a substantial increase in minimum capital requirements over the existing Basel II IRB regulatory capital function.