Cure events in default prediction

This paper evaluates the resurrection event regarding defaulted firms and incorporates observable cure events in the default prediction of SME. Due to the additional cure-related observable data, a completely new information set is applied to predict individual default and cure events. This is a new approach in credit risk that, to our knowledge, has not been followed yet. Different firm-specific and macroeconomic default and cure-event-influencing risk drivers are identified. The significant variables allow a firm-specific default risk evaluation combined with an individual risk reducing cure probability. The identification and incorporation of cure-relevant factors in the default risk framework enable lenders to support the complete resurrection of a firm in the case of its default and hence reduce the default risk itself. The estimations are developed with a database that contains 5930 mostly small and medium-sized German firms and a total of more than 23000 financial statements over a time horizon from January 2002 to December 2007. Due to the significant influence on the default risk probability as well as the bank’s possible profit prospects concerning a cured firm, it seems essential for risk management to incorporate the additional cure information into credit risk evaluation.     

The loss given default of a low-default portfolio with weak contagion

In this paper we study the loss given default (LGD) of a low default portfolio (LDP), assuming that there is weak credit contagion among the obligors. We characterize the credit contagion by a Sarmanov dependence structure of the risk factors that drive the obligors’ default, where the risk factors are assumed to be heavy tailed. From a new perspective of asymptotic analysis, we derive a limiting distribution for the LGD. As a consequence, an approximation for the entire distribution, in contrast to just the tail behavior, of the LGD is obtained. We show numerical examples to demonstrate the limiting distribution. We also discuss possible applications of the limiting distribution to the calculation of moments and the Value at Risk (VaR) of the LGD.     

Optimal consumption and investment with insurer default risk

We solve the optimal consumption and investment problem in an incomplete market, where borrowing constraints and insurer default risk are considered jointly. We derive in closed-form the optimal consumption and investment strategies. We find two main results by quantitative analysis. As insurer default risk increases, the proportion of wealth invested in stocks could increase when wealth is small, and decrease when wealth is large. As risk aversion increases, the voluntary annuity demand could increase when insurer default risk is low, and decrease when this risk is high.     

A reduced-form model with default intensities containing contagion and regime-switching Vasicek processes

The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps (CDSs). Numerical results are presented to show the impacts of the model parameters on the fair spreads.     

A limit distribution of credit portfolio losses with low default probabilities

This paper employs a multivariate extreme value theory (EVT) approach to study the limit distribution of the loss of a general credit portfolio with low default probabilities. A latent variable model is employed to quantify the credit portfolio loss, where both heavy tails and tail dependence of the latent variables are realized via a multivariate regular variation (MRV) structure. An approximation formula to implement our main result numerically is obtained. Intensive simulation experiments are conducted, showing that this approximation formula is accurate for relatively small default probabilities, and that our approach is superior to a copula-based approach in reducing model risk.     

Portfolio problems stopping at first hitting time with application to default risk

In this paper a portfolio problem is considered where trading in the risky asset is stopped if a state process hits a predefined barrier. This state process need not to be perfectly correlated with the risky asset. We give a representation result for the value function and provide a verification theorem. As an application, we explicitly solve the problem by assuming that the state process is an arithmetic Brownian motion. Then the result is used as a starting point to solve and analyze a portfolio problem with default risk modeled by the Black-Cox approach. Finally, we discuss how our results can be applied to a portfolio problem with stochastic interest rates and default risk modeled by the approach of Briys and de Varenne.     

Bayesian confidence intervals for probability of default and asset correlation of portfolio credit risk

We derive Bayesian confidence intervals for the probability of default (PD), asset correlation (Rho), and serial dependence (Theta) for low default portfolios (LDPs). The goal is to reduce the probability of underestimating credit risk in LDPs. We adopt a generalized method of moments with continuous updating to estimate prior distributions for PD and Rho from historical default data. The method is based on a Bayesian approach without expert opinions. A Markov chain Monte Carlo technique, namely, the Gibbs sampler, is also applied. The performance of the estimation results for LDPs validated by Monte Carlo simulations. Empirical studies on Standard & Poor’s historical default data are also conducted.     

Dynamics of multivariate default system in random environment

We consider a multivariate default system where random environmental information is available. We study the dynamics of the system in a general setting of enlargement of filtrations and adopt the point of view of change of probability measures. We also make a link with the density approach in the credit risk modelling. Finally, we present a martingale characterization result with respect to the observable information filtration on the market.     

Predicting default of a small business using different definitions of financial distress

The paper introduces a number of risk-rating models for UK small businesses applying an accounting-based approach, which uses financial ratios to predict corporate bankruptcy. An enhancement to these models is considered through features typical to retail credit risk modelling. A common problem of default prediction consists in the relatively small number of bankruptcies or real defaults available for model-building. In order to expand the ‘default’ group beyond bankrupt companies, the paper considers adopting four different definitions of ‘a failing business’ by investigating combinations of financial distress levels. The impact of each default definition on the choice of predictor variables and on the model's predictive accuracy is explored. In addition, the paper examines the value of categorizing financial ratios used as predictor variables.     

Hurdle models of loan default

Some models of loan default are binary, simply modelling the probability of default, while others go further and model the extent of default (eg number of outstanding payments; amount of arrears). The double-hurdle model, originally due to Cragg (Econometrica, 1971), and conventionally applied to household consumption or labour supply decisions, contains two equations, one which determines whether or not a customer is a potential defaulter (the ‘first hurdle’), and the other which determines the extent of default. In separating these two processes, the model recognizes that there exists a subset of the observed non-defaulters who would never default whatever their circumstances. A Box-Cox transformation applied to the dependent variable is a useful generalization to the model. Estimation is relatively easy using the Maximum Likelihood routine available in STATA. The model is applied to a sample of 2515 loan applicants for whom loans were approved, a sizeable proportion of whom defaulted in varying degrees. The dependent variables used are amount in arrears and number of days in arrears. The value of the hurdle approach is confirmed by finding that certain key explanatory variables have very different effects between the two equations. Most notably, the effect of loan amount is strongly positive on arrears, while being U-shaped on the probability of default. The former effect is seriously under-estimated when the first hurdle is ignored.     

Calibration of the default probability model

In this paper, we study the calibration problem for the Merton–Vasicek default probability model [Robert Merton, On the pricing of corporate debt: the risk structure of interest rate, Journal of Finance 29 (1974) 449–470]. We derive conditions that guarantee existence and uniqueness of the solution. Using analytical properties of the model, we propose a fast calibration procedure for the conditional default probability model in the integrated market and credit risk framework. Our solution allows one to avoid numerical integration problems as well as problems related to the numerical solution of the nonlinear equations.     

Optimal reinsurance with regulatory initial capital and default risk

In a reinsurance contract, a reinsurer promises to pay the part of the loss faced by an insurer in exchange for receiving a reinsurance premium from the insurer. However, the reinsurer may fail to pay the promised amount when the promised amount exceeds the reinsurer’s solvency. As a seller of a reinsurance contract, the initial capital or reserve of a reinsurer should meet some regulatory requirements. We assume that the initial capital or reserve of a reinsurer is regulated by the value-at-risk (VaR) of its promised indemnity. When the promised indemnity exceeds the total of the reinsurer’s initial capital and the reinsurance premium, the reinsurer may fail to pay the promised amount or default may occur. In the presence of the regulatory initial capital and the counterparty default risk, we investigate optimal reinsurance designs from an insurer’s point of view and derive optimal reinsurance strategies that maximize the expected utility of an insurer’s terminal wealth or minimize the VaR of an insurer’s total retained risk. It turns out that optimal reinsurance strategies in the presence of the regulatory initial capital and the counterparty default risk are different both from optimal reinsurance strategies in the absence of the counterparty default risk and from optimal reinsurance strategies in the presence of the counterparty default risk but without the regulatory initial capital.     

Evaluation and default time for companies with uncertain cash flows

In this study, we propose a modelling framework for evaluating companies financed by random liabilities, such as insurance companies or commercial banks. In this approach, earnings and costs are driven by double exponential jump–diffusion processes and bankruptcy is declared when the income falls below a default threshold, which is proportional to the charges. A change of numeraire, under the Esscher risk neutral measure, is used to reduce the dimension. A closed form expression for the value of equity is obtained in terms of the expected present value operators, with and without disinvestment delay. In both cases, we determine the default threshold that maximizes the shareholder’s equity. Subsequently, the probabilities of default are obtained by inverting the Laplace transform of the bankruptcy time. In numerical applications of the proposed model, we apply a procedure for calibration based on market and accounting data to explain the behaviour of shares for two real-world examples of insurance companies.     

General dynamic term structures under default risk

We consider the problem of modelling the term structure of defaultable bonds, under minimal assumptions on the default time. In particular, we do not assume the existence of a default intensity and we therefore allow for the possibility of default at predictable times. It turns out that this requires the introduction of an additional term in the forward rate approach by Heath et al. (1992). This term is driven by a random measure encoding information about those times where default can happen with positive probability. In this framework, we derive necessary and sufficient conditions for a reference probability measure to be a local martingale measure for the large financial market of credit risky bonds, also considering general recovery schemes.     

Asset proportions in optimal portfolios with dependent default risks

In this note, we consider the dependent default risk model of factor type. The dependence between the returns of assets is driven by default indicators. Sufficient conditions on the dependence structure of default indicators and on the utility function are investigated which enable one to order the optimal amount invested in each asset. We thus complement one result in [Cheung, K.C., Yang, H., 2004. Ordering optimal proportions in the asset allocation problem with dependent default risks. Insurance: Math. Econom. 35, 595-609].     

Pricing model of interest rate swap with a bilateral default risk

Under the foundation of Duffie & Huang (1996) [7], this paper integrates the reduced form model and the structure model for a default risk measure, giving rise to a new pricing model of interest rate swap with a bilateral default risk. This model avoids the shortcomings of ignoring the dynamic movements of the firm’s assets of the reduced form model but adds only a little complexity and simplifies the pricing formula significantly when compared with Li (1998) [10]. With the help of the Crank-Nicholson difference method, we give the numerical solutions of the new model to study the default risk effects on the swap rate. We find that for a one year interest rate swap with the coupon paid per quarter, the variance of the default fixed rate payer decreases from 0.1 to 0.01 only causing about a 1.35%’s increase in the swap rate. This is consistent with previous results.     

Restructuring risk in credit default swaps: An empirical analysis

This paper estimates the price for restructuring risk in the US corporate bond market during 1999–2005. Comparing quotes from default swap (CDS) contracts with a restructuring event and without, we find that the average premium for restructuring risk represents 6%–8% of the swap rate without restructuring. We show that the restructuring premium depends on firm-specific balance-sheet and macroeconomic variables. And, when default swap rates without a restructuring event increase, the increase in restructuring premia is higher for low-credit-quality firms than for high-credit-quality firms. We propose a reduced-form arbitrage-free model for pricing default swaps that explicitly incorporates the distinction between restructuring and default events. A case study illustrating the model’s implementation is provided.     

A pricing model for secondary market yield based floating rate notes subject to default risk

The purpose of this article is to price secondary market yield based floating rate notes (SMY-FRNs) subject to default risk. SMY-FRNs are derivatives on the default-free term structure of interest rates, on the term structures for default-risky credit classes, and on the structure of a determined pool of bonds. The main problem in SMY-FRN pricing (as compared to the pricing of standard interest rate or credit derivatives) is market incompleteness, which makes traditional no-arbitrage pricing by replication fail. In general, SMY-FRNs are subject to two types of default risk. First, the SMY-FRN issuer may go bankrupt (direct default risk). Second, the possibility of the bankruptcy of the issuers in the underlying pool has an influence on the SMY-FRN coupons (indirect default risk). This article is the first one which provides a no-arbitrage pricing model for SMY-FRNs with direct and indirect default risks. It is also the first article applying incomplete market pricing methodology to SMY-FRNs.     

信用违约互换的定价方法

通过对信用违约互换的结构的分析,在Merton的结构化方法框架下,用偏微分方程求出公司的违约概率密度,最后给出信用违约互换的一种定价方法.     

Modelling stylized features in default rates

We propose a stochastic model for the probability of default based on diffusions with given marginal distribution and autocorrelation function. The model tries to capture stylized features observed in historical default rates and is analytically tractable. Estimation procedures and expressions for analysis and prediction are provided. Copyright © 2006 John Wiley & Sons, Ltd.