A classification problem of credit risk rating investigated and solved by optimisation of the ROC curve

Estimation of probability of default has considerable importance in risk management applications where default risk is referred to as credit risk. Basel II (Committee on Banking Supervision) proposes a revision to the international capital accord that implies a more prominent role for internal credit risk assessments based on the determination of default probability of borrowers. In our study, we classify borrower firms into rating classes with respect to their default probability. The classification of firms into rating classes necessitates the finding of threshold values separating the rating classes. We aim at solving two problems: to distinguish the defaults from non-defaults, and to put the firms in an order based on their credit quality and classify them into sub-rating classes. For using a model to obtain the probability of default of each firm, Receiver Operating Characteristics (ROC) analysis is employed to assess the distinction power of our model. In our new functional approach, we optimise the area under the ROC curve for a balanced choice of the thresholds; and we include accuracy of the solution into the program. Thus, a constrained optimisation problem on the area under the curve (or its complement) is carefully modelled, discretised and turned into a penalized sum-of-squares problem of nonlinear regression; we apply the Levenberg–Marquardt algorithm. We present numerical evaluations and their interpretations based on real-world data from firms in the Turkish manufacturing sector. We conclude with a discussion of structural frontiers, parametrical and computational features, and an invitation to future work.     

Optimal ROE loan pricing with or without adverse selection

The authors describe the structural solution of the loan rate as a function of default and response risk that maximizes expected return on equity for a lender's portfolio of risky loans. Under the assumptions of our model, the non-linear differential equation for the optimizing price is found to be separable in transformed financial, response and risk variables. With an end-point condition where default-free borrowers are willing to borrow at loan rates higher than the lender's cost of funds, general solutions are obtained for cases where default probabilities may depend explicitly on the offered loan rate and where adverse selection may or may not be present. For the general solution, we suggest a numerical algorithm that involves the sequential solutions of two separate transcendental equations each one of which depends on parameters of the risk and response scores. For the special case where the borrower's default probability is conditionally independent of loan rate, it is shown that the optimal solution is independent of Basel regulations on equity capital.     

On the parameterization of the CreditRisk model for estimating credit portfolio risk

The CreditRisk⁺ model is one of the industry standards for estimating the credit default risk for a portfolio of credit loans. The natural parameterization of this model requires the default probability to be apportioned using a number of (non-negative) factor loadings. However, in practice only default correlations are often available but not the factor loadings. In this paper we investigate how to deduce the factor loadings from a given set of default correlations. This is a novel approach and it requires the non-negative factorization of a positive semi-definite matrix which is by no means trivial. We also present a numerical optimization algorithm to achieve this.     

基于贝叶斯方法的低违约组合违约率估计

现代信用风险建模的核心是估计违约率,违约率估计是否准确将直接影响信用风险建模的质量。在估计违约率的众多文献中,频率法或logistic回归等统计方法的运用非常广泛,此类统计模型的基础是大样本,它客观上需要最低数量或最优数量的违约数据,而低违约组合(LDP)是指只有很少违约数据甚至没有违约数据的组合,如何估计LDP的违约率、反映违约率的非预期波动是一个值得关注的现实问题。本文针对银行贷款LDP缺乏足够历史违约数据的情况,采用贝叶斯方法估计LDP的违约率,并进一步探讨了根据专家判断或者根据同类银行LDP违约数量的历史数据来确定先验分布的方法。在贝叶斯估计中,通过先验分布的设定,不仅可以实现违约率估计的科学性和合理性,而且可以反映违约的非预期波动,有助于银行实施谨慎稳健的风险管理。     

地方政府债券信用风险缓释工具定价研究

随着地方政府债券发行规模的扩大,地方政府债务的信用风险日益凸出。本研究以企业债信用风险缓释工具的推出为契机,借鉴结构化模型的思路和KMV模型求解违约概率的逻辑,通过Monte Carlo方法模拟地方政府的违约过程,直接测算地方政府的整体违约概率;结合简约化模型的思路测算地方政府债券的具体违约概率,计算信用风险缓释工具的理论价格,从而构建了地方政府债券信用风险缓释工具的混合定价模型。研究发现,以企业债券为标的测算出的模型理论价格与市场报价基本一致,参数的敏感性检验进一步验证了模型的理论自洽性和实证可靠性。上述结论或将为新《预算法》实施过程中地方政府债务的治理与掌控及中国区域性、系统性金融风险的防范提供新思路。     

农村信用社信贷违约识别模型及其应用

构建农村信用社信用风险模型对完善农村金融风险管理体系、提高农村信用社经营管理意义重大.基于还款意愿和还款能力两方面,系统分析了影响农信社贷款债务人违约率的主要因素,在此基础上应用logistic方法建立农信社债务人违约率预测模型,并通过Gini系数对模型区分能力和识别能力进行验证评估.实证结果表明,模型中债务人年龄、所在地区、贷款额所占家庭收入比例、与信用社信贷关系密切程度以及户口状况等因素都表现显著;违约率预测模型在样本内和样本外均有较好的违约识别能力,从而可为农信社放贷前的债务人信用评估、贷款发放和风险管理提供有力参考.     

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.     

Correlated intensity, counter party risks, and dependent mortalities

In this paper we use an intensity-based framework to analyze and compute the correlated default probabilities, both in finance and actuarial sciences, following the idea of “change of measure” initiated by Collin-Dufresne et al. (2004). Our method is based on a representation theorem for joint survival probability among an arbitrary number of defaults, which works particularly effectively for certain types of correlated default models, including the counter-party risk models of Jarrow and Yu (2001) and related problems such as the phenomenon of “flight to quality”. The results are also useful in studying the recently observed dependent mortality for married couples involving spousal bereavement. In particular we study in details a problem of pricing Universal Variable Life (UVL) insurance products. The explicit formulae for the joint-life status and last-survivor status (or equivalently, the probability distribution of first-to-default and last-to-default in a multi-firm setting) enable us to derive the explicit solution to the indifference pricing formula without using any advanced results in partial differential equations.     

Computation and analysis for a constrained entropy optimization problem in finance

In [T. Coleman, C. He, Y. Li, Calibrating volatility function bounds for an uncertain volatility model, Journal of Computational Finance (2006) (submitted for publication)], an entropy minimization formulation has been proposed to calibrate an uncertain volatility option pricing model (UVM) from market bid and ask prices. To avoid potential infeasibility due to numerical error, a quadratic penalty function approach is applied. In this paper, we show that the solution to the quadratic penalty problem can be obtained by minimizing an objective function which can be evaluated via solving a Hamilton–Jacobian–Bellman (HJB) equation. We prove that the implicit finite difference solution of this HJB equation converges to its viscosity solution. In addition, we provide computational examples illustrating accuracy of calibration.     

Extending the intensity model with joint defaults to incorporate the lasting effects from common credit events

This paper studies how the lasting effects of common credit events influence default probability distribution and the prices of multiname credit derivatives. Based on a joint defaults model where common credit events are used to generate simultaneous defaults, we extend the model to allow for their impacts to last for a longer while. The default intensity of each entity is heightened significantly while the impact still has an influence, until some time later when this effect fades away. Incorporating these lasting effects helps to generate higher default correlation, which is more consistent with today's highly correlated financial markets. The proposed model can be either formulated as a Markov chain or implemented by Monte Carlo simulation in order to calculate the default probability distributions and multiname derivatives prices. Our numerical results demonstrate the strong influences from the lasting effects and provide a justification of their incorporation.     

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.     

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本文讨论了信用衍生产品之一的总收益互换的定价问题. 其中涉及到利率风险和违约风险, 本文利用HJM利率模型来刻画利率风险, 并利用强度模型和混合模型对违约风险进行建模. 分别考虑了违约时间与利率无关时总收益互换合约的定价问题, 以及违约时间与利率相关时总收益互换合约的定价问题, 给出了相应的定价模型, 并用蒙特卡罗模拟方法得到定价问题的数值解.     

Correlated intensity,counter party risks,and dependent mortalities

In this paper we use an intensity-based framework to analyze and compute the correlated default probabilities, both in finance and actuarial sciences, following the idea of “change of measure” initiated by Collin-Dufresne et al. (2004). Our method is based on a representation theorem for joint survival probability among an arbitrary number of defaults, which works particularly effectively for certain types of correlated default models, including the counter-party risk models of Jarrow and Yu (2001) and related problems such as the phenomenon of “flight to quality”. The results are also useful in studying the recently observed dependent mortality for married couples involving spousal bereavement. In particular we study in details a problem of pricing Universal Variable Life (UVL) insurance products. The explicit formulae for the joint-life status and last-survivor status (or equivalently, the probability distribution of first-to-default and last-to-default in a multi-firm setting) enable us to derive the explicit solution to the indifference pricing formula without using any advanced results in partial differential equations.     

Structural models in consumer credit

We propose a structural credit risk model for consumer lending using option theory and the concept of the value of the consumer’s reputation. Using Brazilian empirical data and a credit bureau score as proxy for creditworthiness we compare a number of alternative models before suggesting one that leads to a simple analytical solution for the probability of default. We apply the proposed model to portfolios of consumer loans introducing a factor to account for the mean influence of systemic economic factors on individuals. This results in a hybrid structural-reduced-form model. And comparisons are made with the Basel II approach. Our conclusions partially support that approach for modelling the credit risk of portfolios of retail credit.     

基于信用利差与Logistic回归的公司违约概率测算模型与实证研究

用Logistic模型计算公司违约概率在实际应用中存在两个问题:一是在缺乏公司违约记录数据库或违约记录数据库不典型的情况下,无法应用该模型或模型计算结果不准确;二是现有Logistic违约概率模型忽视了不同行业财务指标分布特征的差异性,导致公司违约概率计算结果的准确性降低。针对问题一,本文通过公司债券信用利差计算市场隐含的公司违约概率,在Logistic变换的基础上进一步确定Logistic线性回归的参数,使得公司违约概率的计算结果符合债券市场的实际状况。针对问题二,通过不同行业关键财务指标的单因子方差分析,证实了行业间财务指标的分布特征具有显著性差异,通过拟合优度证实了区分行业建立Logistic违约概率模型可显著提高违约概率测算的准确性。本文Logistic违约概率模型的构建过程如下:通过初选财务指标的相关性分析,删除反映信息重复的财务指标;通过Logistic回归中财务指标系数的显著性检验,删除对违约概率解释能力弱的财务指标;以Logistic回归的拟合优度为标准,选取各样本行业Logistic违约概率模型的关键财务指标,建立了机械设备等5个样本行业的Logistic违约概率模型,为样本内行业公司违约概率的准确测算提供模型与方法。本文的创新与特色:一是在无套利条件下,通过公司债券信用利差计算市场隐含的公司违约概率,并对其进行Logistic变换,作为Logistic线性回归的被解释变量,解决了在缺乏公司违约记录数据情况下Logistic违约概率模型的参数估计问题;二是通过单因子方差分析方法,证实了行业间财务指标的分布特征具有显著性差异,说明应区分行业建立Logistic违约概率模型;三是通过财务指标间的相关分析删除反映信息重复的财务指标,通过财务指标系数的显著性检验删除对公司违约概率解释能力弱的财务指标,保证了Logistic违约概率模型中关键财务指标选取的合理性;四是实证研究结果表明,不同行业的Logistic违约概率模型的关键财务指标不同,同一财务指标的参数也存在显著差异。实证研究结果还表明,区分行业建立Logistic违约概率模型与不区分行业相比,前者可将拟合优度及调整后的拟合优度提高近1倍。本文研究结果对于提高公司违约概率测算的准确性具有重要参考意义,对于商业银行贷款定价、公司债券发行定价、银行信用风险管理具有重要参考意义。     

A Reduced Model with Thinning-Dependence Structure

The class of reduced form models is a very important class of credit risk models, and the modelling of the default dependence structure is essential in the reduced form models. This paper models dependent defaults under a thinning-dependent structure in the reduced form framework. In our tractable model, the joint survival probability for correlated defaults can be derived, and hence the CDS premium rates (with or without counterparty risk) are given in closed form. The numerical result shows that the thinning-dependent structure is effective to model the default dependence.     

Modelling the evolution of credit spreads using the Cox process within the HJM framework: A CDS option pricing model

In this paper a simulation approach for defaultable yield curves is developed within the Heath et al. (1992) framework. The default event is modelled using the Cox process where the stochastic intensity represents the credit spread. The forward credit spread volatility function is affected by the entire credit spread term structure. The paper provides the defaultable bond and credit default swap option price in a probability setting equipped with a subfiltration structure. The Euler–Maruyama stochastic integral approximation and the Monte Carlo method are applied to develop a numerical scheme for pricing. Finally, the antithetic variable technique is used to reduce the variance of credit default swap option prices.     

Mixture cure models in credit scoring: If and when borrowers default

Mixture cure models were originally proposed in medical statistics to model long-term survival of cancer patients in terms of two distinct subpopulations - those that are cured of the event of interest and will never relapse, along with those that are uncured and are susceptible to the event. In the present paper, we introduce mixture cure models to the area of credit scoring, where, similarly to the medical setting, a large proportion of the dataset may not experience the event of interest during the loan term, i.e. default. We estimate a mixture cure model predicting (time to) default on a UK personal loan portfolio, and compare its performance to the Cox proportional hazards method and standard logistic regression. Results for credit scoring at an account level and prediction of the number of defaults at a portfolio level are presented; model performance is evaluated through cross validation on discrimination and calibration measures. Discrimination performance for all three approaches was found to be high and competitive. Calibration performance for the survival approaches was found to be superior to logistic regression for intermediate time intervals and useful for fixed 12 month time horizon estimates, reinforcing the flexibility of survival analysis as both a risk ranking tool and for providing robust estimates of probability of default over time. Furthermore, the mixture cure model’s ability to distinguish between two subpopulations can offer additional insights by estimating the parameters that determine susceptibility to default in addition to parameters that influence time to default of a borrower.     

Efficient calibration for problems in option pricing

The pricing of derivatives has gained considerable importance in the finance industry and leads to challenging problems in numerical optimization. We focus on the numerical solution of a stochastic model for option prices. In particular, we are concerned with the calibration of these models to real data, which leads to large scale optimization problems. We consider the numerical solution of these optimization problems and give some indication how to reduce the complexity of these problems. Special emphasis is devoted to a multi-layer strategy which is embedded into the optimization iteration. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

DEA–DA for bankruptcy-based performance assessment: Misclassification analysis of Japanese construction industry

This study describes a practical use of Data Envelopment Analysis–Discriminant Analysis (DEA–DA) for bankruptcy-based performance assessment. DEA–DA is useful for classifying non-default and default firms based upon their financial performance. However, when we apply DEA–DA to a data set on corporate bankruptcy, we usually face three problems. First, there is a sample imbalance problem because the number of default firms is often limited. In contrast, we can easily obtain a large number of non-default firms. Second, there is a computational problem to deal with a large data set. We need to consider a computational strategy to reduce the dimension of a large data set. Finally, we need to consider data alignment because the location of default firms may exist within that of non-default firms. This study discusses a simultaneous occurrence of the three problems from the perspective of Japanese industrial policy on construction business. To handle the three problems, this study combines DEA–DA with principal component analysis to reduce the computational burden and then alters DEA–DA weights to address both the sample imbalance problem and the location problem. This study also discusses a combined use between DEA–DA and rank sum tests to examine statistically hypotheses related to bankruptcy assessment. As an important application, we apply the proposed approach to the Japanese construction industry and discuss why many Japanese construction firms are misclassified.