Ebook From Fault Tree to Credit Risk Assessment: A Case Study
Whatever the considered matter, risk has become of major interest since the 80' (see Cabarbaye [1998], Henley and Kumamoto [1992], and Papoulis [1984] for example). The risk that a disastrous event occurs is of great importance since such an event engenders social harm as well as economic and financial losses. This principle also applies to credit risk valuation, which has been widely focused since the last decade. Indeed, Basel II directives underline the importance of the ability to value and quantify fairly default risk (see Basel Committee on banking supervision [1996] for example).
Therefore, the sound and reliable assessment of default risk represents the challenge of the next decade. Along with this consideration, we employ the simple setting of Gatfaoui (2006) to value credit risk. The author applies fault tree theory to assess default risk in a simple framework (see Bon [1995] and Rothenthal [1998] among others). More precisely, considering the empirical probabilities that French firms go bankrupt (i.e., empirical default probabilities), fault tree analysis allows the author for estimating related hazard rates, or equivalently, failure rates of these French firms. Indeed, the study focuses on the lifetimes of French firms since any firm s default probability corresponds to the probability of death of the firm, or equivalently, to the probability that its lifetime ends. Therefore, failure rates' estimations depend on the probability distribution of related lifetimes. Gatfaoui (2006) chose to resort to an exponential law with a constant intensity in order to describe French firms' default probabilities.
However, although this statistical representation seems to be appropriate, the author finds that corresponding implied failure rates are time varying, and exhibit a convex decreasing pattern. Such a result is in accordance with the work of Fons and Kimball (1991) who highlight the significant time varying behavior of failure rates. This time varying feature is shown to be as important as firms' credit ratings are low. Moreover, choosing an exponential law with a constant parameter implicitly assumes a time independency for the hazard rate function (i.e., the present does not depend on the past). Such an assumption is nevertheless inconsistent with modern default risk analysis. Indeed, bankruptcy threatens especially young firms under five years old, supporting the existence of a life cycle for firms. Such a consideration suggests a time dependency for the hazard function. Namely, hazard rates or, equivalently, default probabilities should have higher levels at the beginning of newly created firms' existence.
Although a time varying intensity exponential law can be proxied by a series of constant intensity exponential laws over well chosen and sufficiently small time subsets, we focus on the global behavior of hazard rates. We propose consequently to extend the work of Gatfaoui (2006) in order to take into account the time dependency of the failure rate function. For this purpose, we consider the following set of probability distributions, namely lognormal, log-logistic, gamma, Weibull, beta of second species, a mixture of two exponential laws with constant intensity, and finally, two non-homogeneous Poisson processes known as Cox-Lewis and exponential exponent. In this way, we are able to account for a wide range of monotonous, hump-shaped, and convex or concave failure rates relative to time. And, we can capture most of empirical well known patterns describing corporate failures. We hope our framework to allow for parameter estimates yielding defined and finite mean and variance for distributions. Indeed, the existence of the two first moments is extremely important in characterizing reliability (i.e., survival time of firms). Specifically, our distribution set requires the existence and boundedness of their respective mean and variance (i.e., volatility). These two moments belong to the key parameters and conditions that define each of our eight possible probability distributions in a theoretical viewpoint (e.g., mean time to failure).
Our paper is organized as follows. First, we recall the theoretical reliability framework (i.e., basic notions and principles) and the characteristics of each probability distribution (i.e., statistical properties). Second, we present related results (e.g., parameter estimates for each distribution-type) and the Kolmogorov adequacy test ensuring the soundness of our representations. We also perform an exponentiality test to investigate the coherency of our non-homogeneous Poisson processes versus the classic exponential law with constant intensity while describing French failures. Such a test allows us to investigate the usefulness and relevance of a time varying intensity parameter versus a constant one in exponential-type representations. Third, we look for the optimal representation of our failure rates given the set of consistent probability distributions, and compute related forward conditional default probabilities over various time horizons.
The optimality criterion we employ solves a quadratic problem, namely the minimization of some absolute error function. Hence, the optimal characterization fits at best the empirical default probabilities under consideration. Fourth, we use the obtained optimal representations to deduce the corresponding term structure of credit risky discount bonds. By the way, we underline the link prevailing between the reduced form approach of credit risk and reliability.
Precisely, the reduced form approach is known to often stipulate a priori dynamics for risky bonds' term structure and therefore credit spreads' term structure. This branch of credit risk assessment is based on the default time s arrival. Specifically, default time is represented as a random variable since the instant of potential default is unknown and uncertain. Such a setting is therefore founded on the intensity process of default, which describes the probability that a default event occurs over any infinitesimal time interval. Hence, characterizing credit spreads' term structure requires only information about both the hazard rate function and the corresponding potential recovery rate (when the risk free term structure is deterministic at most).
Finally, we end our paper with some concluding remarks and possible extensions to our analysis in the lens of time dependency and business cycle s impact. Specifically, economic world s changes impact default risk (i.e., possibility of corporate failures at any time) as time elapses. Recall that any risk profile is defined by two main dimensions, namely time and uncertainty.
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