1、 外文翻译 原文 Modeling consumer credit risk via survival analysis Material Source: SORT33(1)January-June 2009， 3-30 Author: Ricardo Cao. Juan M. Velar and Andres Devi 1. Abstract Credit risk models are used by financial companies to evaluate in advance the insolvency risk caused by credits that enter int
2、o default. Many models for credit risk have been developed over the past few decades. In this paper, we focus on those models that can be formulated in terms of the probability of default by using survival analysis techniques. With these objective three different mechanisms are proposed based on the
3、 key idea of writing the default probability in terms of the conditional distribution function of the time to default. The first method is based on a Coxs regression model, the second approach uses generalized linear models under censoring and the third one is based on nonparametric kernel estimatio
4、n, using the product-limit conditional distribution function estimator by Bearn. The resulting nonparametric estimator of the default probability is proved to be consistent and asymptotically normal. An empirical study, based on modified real data, illustrates the three methods. 2. Introduction Dete
5、rmining the probability of default, PD, in consumer credits, loans and credit cards is one of the main problems to be addressed by banks, savings banks, savings cooperatives and other credit companies. This is a first step needed to compute the capital in risk of insolvency, when their clients do no
6、t pay their credits, which is called default. The risk coming from this type of situation is called credit risk, which has been the object of research since the middle of last century. The importance of credit risk, as part of financial risk analysis, comes from the New Basel Capital Accord (Basel I
7、I), published in 1999 and revised in 2004 by the Basel Committee for Banking Supervision (BCBS). This accord consists of three parts, called pillars. They constitute a universal theoretical framework for the procedures to be followed by credit companies in order to guarantee minimal capital requirem
8、ents, called statistical provisions for insolvency (SPI). Pillar I of the new accord establishes the parameters that play some role in the credit risk of a financial company. These are the probability of default, PD, the exposition after default, EAD, and the loss given default, LGD. The quantitativ
9、e methods that financial entities can use are those used for computing credit risk parameters and, more specifically, for computing PD. These are the standard method and the internal ratings based method (IRB). Thus, credit companies can elaborate and use their own credit qualification models and, b
10、y means of them, conclude the Basel implementation process, with their own estimations of SPI. There is an extensive literature on quantitative methods for credit risk, since the classical Z-score model introduced by Altman (1968). Nowadays there exist plenty of approaches and perspectives for model
11、ing credit risk starting from PD. Most of them have provided better predictive powers and classification error rates than Altmans discriminant model, for credit solicitors (application scoring), as well as for those who are already clients of the bank (behavioral scoring). This is the case of logist
12、ic regression models; artificial neural networks (ANN), support vector machines (SVM), as well as hybrid models, as mixtures of parametric models and SVM. For the reader interested in a more extended discussion on the evolution of these techniques over the past 30 years we mention the work by Altman
13、 and Saunders (1998), Saunders (1999), Croupy et al. (2000), Hand (2001), Hamerle et al. (2003), Hanson and Sherman (2004),Wang et al.(2005),and Chant al.(2006). The main aim of this paper is to introduce an alternative approach for modeling credit risk. More specifically, we will focus on estimatin
14、g PD for consumer credits and personal credits using survival analysis techniques. The idea of using survival analysis techniques for constructing credit risk models is not new. It started with the paper by Marian (1992) and, later, was developed by Carling et al. (1998), Stepanova and Thomas (2002)
15、, Roszbach (2003), Guenon and Negron (2005), Allen and Rose (2006), Baba and Gobo (2006), Manlike and Thomas (2006) and Bearn and Jidda (2007). A common feature of all these papers is that they use parametric or semi parametric regression techniques for modeling the time to default (duration models)
16、, including exponential models, Waybill models and Coxs proportional hazards models, which are very common in this literature. The mode established for the time to default is then used for modeling PD or constructing the scoring discriminate function. In this paper we propose a basic idea to estimat
17、e PD, which is performed by three different methods. The first one is based on Coxs proportional hazards model, the second one uses generalized linear models, while the third one consists in using a random design nonparametric regression model. In all the cases, some random right censoring mechanism
18、 appears in the model, so survival analysis techniques are natural tools to be used The conditional survival function used for modeling credit risk opens an interesting perspective to study default. Rather than looking at default or not, we look at the time to default, given credit information of cl
19、ients (endogenous covariates) and considering the indicators for the economic cycle (exogenous covariates). Thus, the default risk is measured via the conditional distribution of the random variable time to default, T, given a vector of covariates, X. The variable T is not fully observable due to th
20、e censoring mechanism. In order to estimate empirically the conditional distribution function of the time to default, we use the generalized product-limit estimator by Bearn(1981).This estimator has been extensively studied by Dabrowska (1987), Dabrowska (1989), Gonzalez- Mantegna and Cedars-Suarez
21、(1994),Van Kialegee and Veraverbeke (1996),Iglesias-Perez and Gonzalez-Mantegna(1999), Li and Datta (2001), Van Kialegee et al (2001) and Li and Van Kialegee (2002), among other authors. The rest of the paper proceeds as follows. Section 2 presents some conditional functions, often used in survival
22、analysis, and explains how they can be used for credit risk analysis. The estimation of the probability of default is considered in Section 3, under different models: Coxs proportional hazards model, generalized linear models and a nonparametric model. Special attention is given to the study of the
23、theoretical NPM properties of the nonparametric estimator for PD, denoted by PDNMMP. Its asymptotic bias and variance, as well as uniform consistency and asymptotic normality are stated in Section 4. An application to a real data set, with a brief discussion about the empirical results obtained, is
24、presented in Section 5. Finally, Section 6 contains the proofs of the results included in Section4. 3. Conditional survival analyses in credit risk The use of survival analysis techniques to study credit risk, and more particularly to model PD, can be motivated via Figure1. It presents three common
25、situations that may occur in practice when a credit company observes the “life time” of a credit. Figure 1: Time to default in consumer credit risk. 4. Results for the generalized linear model Figure 5 show the results obtained for the PD estimated with the GLM model using two parametric links: Pare
26、to and Seducers F. The range of the estimated PD lies within the interval 0, 0.016 when the link function is Pareto and grows up to the interval 0, 0.378 when the link function is F(10,50), as it can be seen in Table 2. The PD curves obtained with this model are exponentially decreasing, as expected
27、, but in this case it seems that there is no a significantly contribution of the variable X in the accuracy of the estimated default probability curves. Furthermore, the estimated curves are all above the range of the observed default rate with maturity one year forward. The results achieved by usin
28、g these two parametric links do not fit as well as expected to the data, when compared to the empirical default rate curves depicted in Figure 3. In spite of this, the GLM method may be useful to study the PD horizon in the long run. Other link distributions belonging to the exponential family have
29、been used to fit these data via GLM. The normal distribution, the Waybill distribution and the Cauchy distribution were used, among others. The results obtained were even worse than those presented in Figure 5 above. 5. Probability of default in consumer portfolio In the literature devoted to credit
30、 risk analysis there are not many publications on modeling the credit risk in consumer portfolios or personal credit portfolios. Most of the research deals with measuring credit risk by PD modeling in portfolios of small, medium and large companies, or even for financial companies. There exist, howe
31、ver, several exceptions. In the works by Carling et al. (1998), Stepanova and Thomas (2002) and Manlike and Thomas (2006), the lifetime of a credit is modeled with a semi parametric regression model, more specifically with Coxs proportional hazards model. In the following we present three different
32、approaches to model the probability of default, PD, using conditional survival analysis. All the models are based on writing PD in terms of the conditional distribution function of the time to default. Thus PD can be estimated, using this formula, either by (i) Coxs proportional hazards model, where
33、 the estimator of the survival function is obtained by solving the partial likelihood equations PHM in Coxs regression model, which gives PD ,by(ii) a generalized linear model, with GLM parameters estimated by the maximum likelihood method, which gives PD, or by(iii) using the nonparametric conditio
34、nal distribution function estimator by Bearn, which NPM gives the nonparametric estimator of the default probability, PD. 6. Comparisons A summary with a descriptive comparison of the three models is given in Table2. Fixed values for the covariate X (first, second and third quartiles) were used for
35、the conditional distributions. Of course, the empirical default rate does not depend on the value of X. Although no goodness-of-fit tests have been applied for the proposed models, the results of the estimation can be checked by simple inspection of Figures 47 and the descriptive statistics collecte
36、d in Table 2. The results for each model can be compared with those of the aggregated default rates in the whole portfolio. Such values should be considered as a reference value for the three models. 译文 消费信贷风险建模生存分析 资料来源： SORT33(1)January-June 2009， 3-30 作者：里卡多 曹和安德烈 1 摘要 许多金融公司使用信用风险模型来提前评估因拖欠的资金而引
37、起的破产的风险。在过去的几十年里许多信用风险模型已经有了发展。在本文中，我们关注的是那些通过使用生存分析技术来制定的有关资金拖欠可能性方面的模型。在条件分布函数上随着时间推移违约可能性为关键的基础上，三种客观存在的机制被提出。第一种方法是基于考克斯的回归模型 ,第二种方法采用广义线性模型下的审查，第三种是基于非参数 核估计的基础上，由 Bearn 运用 product-limit条件分布函数估计。由此产生的非参数估计违约概率被证实是一致的和渐近正常。在真实数据的基础上，一项经验学的研究说明了这三种方法。 2 介绍 确定在消费信贷、贷款和使用信用卡违约的可能性、 客户违约概率 是现在银行 、 储
38、蓄银行和储蓄合作社和其他信贷公司需要解决的一个重要问题。当其客户不支付其贷款，即我们所说的违约时，这是计算资金无力偿还风险所需的第一步。这种风险来自于被称作是信用风险的情境，它从上个世纪中期以来就已经成为了研究的对象。作为财务风险分析 的一部分，信用风险的重要性来自于塞尔新资本协议 （ 新巴塞尔协议 ） ，发表于 1999 年和在 2004 巴塞尔委员会为银行业监督管理 （ BCBS） 进行了修订。该协议是由三部分组成 ,称为柱子。他们组成了 现在 许多金融公司为了确保最低限度的资本要求所遵循的普遍理论框架，即破产的统计学规定。 协议中的支柱一：建立在金融企业的信贷危机中扮演一定角色的参数。他
39、们是违约的可能性 PD，违约后的暴露 EAD，违约带来的损失 LGDA， 金融公司可以使用的定量方法是 哪 些用来计算信贷风险参数，更确切的说，是计算 PD。这些都是标 准法和内部基础评级法 （ IRB） ，因此 ， 信贷公司可以生产和使用他们自己的信用资格模型 ， 而且通过这些模型，以对 SPI 的评估总结出巴塞尔实施的过程。 自从由 Altman（ 1968）介绍经典的 Z-score 模型后，出现了大量关于信贷风险计量方法的文献。从 客户违约概率（ PD） 开始， 现在有许多 用于建立信贷风险模型的方法和观点。他们中的大多数可以为信贷律师和那些银行的顾客提供比 Altman 的分类模型更
40、好的预测能力和分类错误比率。这是在逻辑回归模型的情况下，人工神经网络 (ANN)， 支持向量机 (SVM)以及像参数化模 型和 SUM的混合模型。为读者感兴趣的是关于过去的 30 年里这些技术 进步的一场大讨论，我们要提到 Altman and Saunders 的作品（ 1998）， Saunders (1999)， Croupy的作品 (2000), Hand (2001)和 Homer 的作品 (2003)， Hanson and Sherman 的作品（ 2004）， Wang 的作品 (2005),and Chant 的作品 (2006)。 本文的主要目的是介绍一种可选择的方法 来
41、建立信贷风险模型。更具体地说 是 ，我们将集中于利用生存分析统计技术 来评估消费者和个人信贷的 客户违约概率 。 这种利用生存分析统计技术建立信用风险模型并不是新的想法。它由Marian（ 1992） 的论文开始，然后由 Carling 的论文 (1998)， Stepnova and Thomas (2002),Roszbach (2003),Guenon and Negron(2005), Allen and Rose (2006), Baba and Goon (2006), Manlike and Thomas (2006) and Bearn and Jidda (2007)来发展
42、的 。所有这些论文的共同点是他们都 是 使用参数或半参数回归分析方法来建立随时间而违约的模型（ 即 时长模型）。包括指数模型 ， 威布尔模型和在这个领域里面很普遍的考克斯的相对危险模型。这些时长模型后来被用作建立 客户违约概率 模型或者用来评价判别函数。 在本文中 ，我们提出一个基本理念来估计 客户违约概率， 它 是 以三种不同的形式表现出来 的 。第一种是基于考克斯的相对危险模型 来说的， 第二个 是 采用广义 的 线性模型 ， 而第三个是使用一个随机设计 的非参数回归模型。在所有的案例中，一些随机的正确审查机制出现在这些 模型中，生存分析技术是 作为一个 常用的工具来使用 的 。 用来建立
43、信贷风险模型的有条件的生存函数开创了一个有趣的角度 来 研究违约拖欠。在给了客户的信贷信息和考虑到经济周期后，我们与其看客户拖欠与否，不如看 违约期限 。因此， 违约风险是通过随便变量的违约时间分布来测量的。 给定一个向量的趋势 X。 变量 t 不能完全 被 观察到 的是因为审查机制 。 为了估计 违约时间 的条件分布函数，我们使用 Bearn (1981)的广义product-limit 评价人。这个评价人被 Dabrowska(1987),Dabrowska(1989), Gonzalez-Mantegna and Cedars-Suarez(1994),Van Kialegee and
44、Veraverbeke (1996),Iglesias-Perez and Gonzalez-Mantegna(1999),Li and Datta(2001),Van Kialegee(2001)and Li and Van Kialegee(2002),和其他作者研究过。 这篇文章剩下的部分如下： 第二部分呈现了一些条件函数，它们经常在生存分析中被用到，也用 来解释它们怎样被用作信贷风险 来 分析。第三部分考虑了违约的可能性，有以下三种模式：考克斯的相对危险模型、广义 的 线性模型和非参数模型。特别注意了客户违约概率 的非参数评估 NPM 属 性理论上的研究。第四节陈述了其渐近估计偏差与方
45、差性能 ， 以及均匀一致性和渐近正态性。第五部分呈现了真实数据的应用，以及对实证结果进行了一个小讨论。最后，第六部分包含了前述的，包括第四部分的结果。 2 条件生存分析在信贷风险中的应用 利用生存分析方法来研究信用风险，更确切的说，研究 客户违约风险 模式，可以 通过图 1 被 激发。当金融公司观察信贷的“ 生活”时可能会出现 ， 它提出了三种常见的情境 。 图 1 在消费者信贷风险中的违约时间 3 广义 的线性模型 成果 现实的结果包含了 GLM 模型应用两个参数连接来评估 客户违约概率 ：Pareto and Seducer。客户违约概率 的估计范围在区间 0， 0.016，当连接函数的参
46、数是 F(10， 50)，就像在表 2 中看到的那样， Pareto 的连接函数的区间有所增高，客户违约概率的评计范围在区间 0,0.378。在该模型中，正如预期的那样 客户违约概率 曲线是按指数衰减的，可是在这个案例中，没有显著表明变量 X 和估计违约概率曲线 之间的明确性。此外，估计曲线都在违约率的范围之上。当和表 3 中的实证违约率曲线相比，这个通过两个参数连接获得的结果不适合用于预期的数据。尽管如此， GLM 方法可能有助于研究 PD视界的长远发展。其他属于指数族的联合分布已经被用于分析通过 GLM 获得的数据。另外，正态分布、威布尔分布和柯西也使用着。他们的结果甚至比上表 5 中的更
47、坏。 4 消费者违约的概率组合 目前在致力于信贷风险分析的文献中，很少有关于建立关于消费信贷和个人信贷的投资组合的刊物。大多数的研究是通过使用小，中，大公司的组合 PD模型来处理衡量信贷风险，或者 甚至是金融公司。但是这里存在着几个例外。在 Carling (1998), Stepanova and Thomas(2002) and Manlike and Thomas(2006)的作品中，一生中一个信用进行建模与半参数回归模型 ， 更确切的说 是 考克斯的相对危险模型。 在接下来，我们提出了三种不同的模型通过使用条件生存分析来衡量违约拖欠的可能性 PD。所有的模型都是基于 违约时间 的条件分
48、布函数方面的。所以运用该公式，无论是使用考克斯的相对危险模型的估计 ， 在生存 功能 归纳为求解局部似然方程 PHM 在考克 斯的回归模型 ,， 这就有了 客户违约概率 ；或者是用广义 的 线性模型，用单因数变异数参数的极大似然法估计，这就有了 客户违约概率 ；再或者是利用 Bearn 的非参数估计的条件分布函数， NPM 给了违约可能性的非参数估计。 5 比较 表 2 中给出了三种模式的总结和描述性的比较 ， 共向变量 X 的固定值（第一、第二和第三的四分之一）被用作条件分布。当然实证违约率不依赖于 x 的数值。 虽然没有拟合优度试验已经申请建立模型，表格 4-7 的简单检查和表格 2的描述性数据收集来检测估计的结果。每个模式的结果可以在整个投资组合中和 上年度违约率比较。这样的价值应被视为参考价值的三种模式。
