Time to default in credit scoring using survival analysis: a benchmark study

We investigate the performance of various survival analysis techniques applied to ten actual credit data sets from Belgian and UK financial institutions. In the comparison we consider classical survival analysis techniques, namely the accelerated failure time models and Cox proportional hazards regression models, as well as Cox proportional hazards regression models with splines in the hazard function. Mixture cure models for single and multiple events were more recently introduced in the credit risk context. The performance of these models is evaluated using both a statistical evaluation and an economic approach through the use of annuity theory. It is found that spline-based methods and the single event mixture cure model perform well in the credit risk context.     

Neural network survival analysis for personal loan data

Traditionally, credit scoring aimed at distinguishing good payers from bad payers at the time of the application. The timing when customers default is also interesting to investigate since it can provide the bank with the ability to do profit scoring. Analysing when customers default is typically tackled using survival analysis. In this paper, we discuss and contrast statistical and neural network approaches for survival analysis. Compared to the proportional hazards model, neural networks may offer an interesting alternative because of their universal approximation property and the fact that no baseline hazard assumption is needed. Several neural network survival analysis models are discussed and evaluated according to their way of dealing with censored observations, time-varying inputs, the monotonicity of the generated survival curves and their scalability. In the experimental part, we contrast the performance of a neural network survival analysis model with that of the proportional hazards model for predicting both loan default and early repayment using data from a UK financial institution.     

PHAB scores: proportional hazards analysis behavioural scores

Credit scoring is one of the most widely used applications of quantitative analysis in business. Behavioural scoring is a type of credit scoring that is performed on existing customers to assist lenders in decisions like increasing the balance or promoting new products. This paper shows how using survival analysis tools from reliability and maintenance modelling, specifically Cox's proportional hazards regression, allows one to build behavioural scoring models. Their performance is compared with that of logistic regression. Also the advantages of using survival analysis techniques in building scorecards are illustrated by estimating the expected profit from personal loans. This cannot be done using the existing risk behavioural systems.     

Cox模型与BP神经网络在处理非线性数据时的性能比较

本文采用BP神经网络、Cox模型和bootstrap方法,比较BP神经网络与Cox模型在处理非线性资料时的性能。两种方法的预测一致性的均数分别为0.7525和0.7706。对于非线性资料,BP神经网络的预测效果优于Cox模型。     

A time-dependent proportional hazards survival model for credit risk analysis

In the consumer credit industry, assessment of default risk is critically important for the financial health of both the lender and the borrower. Methods for predicting risk for an applicant using credit bureau and application data, typically based on logistic regression or survival analysis, are universally employed by credit card companies. Because of the manner in which the predictive models are fit using large historical sets of existing customer data that extend over many years, default trends, anomalies, and other temporal phenomena that result from dynamic economic conditions are not brought to light. We introduce a modification of the proportional hazards survival model that includes a time-dependency mechanism for capturing temporal phenomena, and we develop a maximum likelihood algorithm for fitting the model. Using a very large, real data set, we demonstrate that incorporating the time dependency can provide more accurate risk scoring, as well as important insight into dynamic market effects that can inform and enhance related decision making.     

The benefit to consumers from generic scoring models based on credit reports

Received on 1 July 1991. The benefit to consumers from the use of informative creditreports is demonstrated by showing the improvement in creditdecisions when generic scoring models based on credit reportsare implemented. If these models are highly predictive, thenthe truncation of credit reports will reduce the predictivepower of bureau-based generic scoring systems. As a result,more good credit risks will be denied credit, and more poorcredit risks will be granted credit. It is shown that, evenwhen applied to credit applications that had already been screenedand approved, the use of generic scoring models significantlyimproves credit grantors' ability to predict and eliminate bankruptcies,charge-offs, and delinquencies. As applied to existing accounts,bureau-based generic scores are shown to have predictive valuefor at least 3 months, while scores 12 months old may not bevery powerful. Even though bureau-based scores shift towardsthe high-risk end of the distribution during a recession, theycontinue to rank risk very well. When coupled with application-basedcredit-scoring models, scores based on credit-bureau data furtherimprove the predictive power of the model-the improvements beinggreater with more complete bureau information. We conclude thatgovernment-imposed limits on credit information are anti-consumerby fostering more errors in credit decisions.     

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.     

A comparison of neural networks and linear scoring models in the credit union environment

The purpose of the present paper is to explore the ability of neural networks such as multilayer perceptrons and modular neural networks, and traditional techniques such as linear discriminant analysis and logistic regression, in building credit scoring models in the credit union environment. Also, since funding and small sample size often preclude the use of customized credit scoring models at small credit unions, we investigate the performance of generic models and compare them with customized models. Our results indicate that customized neural networks offer a very promising avenue if the measure of performance is percentage of bad loans correctly classified. However, if the measure of performance is percentage of good and bad loans correctly classified, logistic regression models are comparable to the neural networks approach. The performance of generic models was not as good as the customized models, particularly when it came to correctly classifying bad loans. Although we found significant differences in the results for the three credit unions, our modular neural network could not accommodate these differences, indicating that more innovative architectures might be necessary for building effective generic models.     

Data mining feature selection for credit scoring models

The features used may have an important effect on the performance of credit scoring models. The process of choosing the best set of features for credit scoring models is usually unsystematic and dominated by somewhat arbitrary trial. This paper presents an empirical study of four machine learning feature selection methods. These methods provide an automatic data mining technique for reducing the feature space. The study illustrates how four feature selection methods—‘ReliefF’, ‘Correlation-based’, ‘Consistency-based’ and ‘Wrapper’ algorithms help to improve three aspects of the performance of scoring models: model simplicity, model speed and model accuracy. The experiments are conducted on real data sets using four classification algorithms—‘model tree (M5)’, ‘neural network (multi-layer perceptron with back-propagation)’, ‘logistic regression’, and ‘k-nearest-neighbours’.     

A profit-based scoring system in consumer credit: making acquisition decisions for credit cards

Consumer credit scoring is one of the most successful applications of quantitative analysis in business with nearly every major lender using charge-off models to make decisions. Yet banks do not extend credit to control charge-off, but to secure profit. So, while charge-off models work well in rank-ordering the loan default costs associated with lending and are ubiquitous throughout the industry, the equivalent models on the revenue side are not being used despite the need. This paper outlines a profit-based scoring system for credit cards to be used for acquisition decisions by addressing three issues. First, the paper explains why credit card profit models—as opposed to cost or charge-off models—have been difficult to build and implement. Second, a methodology for modelling revenue on credit cards at application is proposed. Finally, acquisition strategies are explored that use both a spend model and a charge-off model to balance tradeoffs between charge-off, revenue, and volume.     

Support vector machines for classifying and describing credit applicants: detecting typical and critical regions

Credit applicants are assigned to good or bad risk classes according to their record of defaulting. Each applicant is described by a high-dimensional input vector of situational characteristics and by an associated class label. A statistical model, which maps the inputs to the labels, can decide whether a new credit applicant should be accepted or rejected, by predicting the class label given the new inputs. Support vector machines (SVM) from statistical learning theory can build such models from the data, requiring extremely weak prior assumptions about the model structure. Furthermore, SVM divide a set of labelled credit applicants into subsets of ‘typical’ and ‘critical’ patterns. The correct class label of a typical pattern is usually very easy to predict, even with linear classification methods. Such patterns do not contain much information about the classification boundary. The critical patterns (the support vectors) contain the less trivial training examples. For instance, linear discriminant analysis with prior training subset selection via SVM also leads to improved generalization. Using non-linear SVM, more ‘surprising’ critical regions may be detected, but owing to the relative sparseness of the data, this potential seems to be limited in credit scoring practice.     

On reduced-form intensity-based model with ‘trigger’ events

Corporate defaults may be triggered by some major market news or events such as financial crises or collapses of major banks or financial institutions. With a view to develop a more realistic model for credit risk analysis, we introduce a new type of reduced-form intensity-based model that can incorporate the impacts of both observable ‘trigger’ events and economic environment on corporate defaults. The key idea of the model is to augment a Cox process with ‘trigger’ events. Both single-default and multiple-default cases are considered in this paper. In the former case, a simple expression for the distribution of the default time is obtained. Applications of the proposed model to price defaultable bonds and multi-name Credit Default Swaps are provided.     

Towards profitability: a utility approach to the credit scoring problem

Since credit scoring was first applied in the 1940s the standard methodology has been to treat consumer lending decisions as binary classification problems, where the goal has been to make the best possible ‘good/bad’ classification of accounts on the basis of their eventual delinquency status. However, the real goal of commercial lending organizations is to forecast continuous financial measures such as contribution to profit, but there has been little research in this area. In this paper, continuous models of customer worth are compared to binary models of customer repayment behaviour. Empirical results show that while models of customer worth do not perform well in terms of classifying accounts by their good/bad status, they significantly outperform standard classification methodologies when ranking accounts based on their financial worth to lenders.     

Improving credit scoring by differentiating defaulter behaviour

We present a methodology for improving credit scoring models by distinguishing two forms of rational behaviour of loan defaulters. It is common knowledge among practitioners that there are two types of defaulters, those who do not pay because of cash flow problems (‘Can’t Pay’), and those that do not pay because of lack of willingness to pay (‘Won’t Pay’). This work proposes to differentiate them using a game theory model that describes their behaviour. This separation of behaviours is represented by a set of constraints that form part of a semi-supervised constrained clustering algorithm, constructing a new target variable summarizing relevant future information. Within this approach the results of several supervised models are benchmarked, in which the models deliver the probability of belonging to one of these three new classes (good payers, ‘Can’t Pays’, and ‘Won’t Pays’). The process improves classification accuracy significantly, and delivers strong insights regarding the behaviour of defaulters.     

Reject inference in survival analysis by augmentation

The literature suggests that the commonly used augmentation method of reject inference achieves no appreciable benefit in the context of logistic and probit regression models. Ranking is not improved and the ability to discern a correct cut-off is undermined. This paper considers the application of augmentation to profit scoring applicants by means of survival analysis and by the Cox proportional hazard model, in particular. This new context involves more elaborate models answering more specific questions such as when will default occur and what will be its precise financial implication. Also considered in this paper is the extent to which the rejection rate is critical in the potential usefulness of reject inference and how augmentation meets that potential. The conclusion is essentially that augmentation achieves negative benefits only and that the scope for reject inference in this context pertains mainly to circumstances where a high proportion of applicants have been rejected.     

SICA for Cox’s proportional hazards model with a diverging number of parameters

The smooth integration of counting and absolute deviation （SICA） penalized variable selection procedure for high-dimensional linear regression models is proposed by Lv and Fan （2009）. In this article, we extend their idea to Cox＇s proportional hazards （PH） model by using a penalized log partial likelihood with the SICA penalty. The number of the regression coefficients is allowed to grow with the sample size. Based on an approximation to the inverse of the Hessian matrix, the proposed method can be easily carried out with the smoothing quasi-Newton （SQN） algorithm. Under appropriate sparsity conditions, we show that the resulting estimator of the regression coefficients possesses the oracle property. We perform an extensive simulation study to compare our approach with other methods and illustrate it on a well known PBC data for predicting survival from risk factors.     

The flat-maximum effect and generic linear scoring models: a test

Received on 1 July 1991. Predicting human behaviour patterns with linear correlationmodels has absorbed researchers for the past five decades. Althoughmost observers generally concede that humans are inferior tosuch models in combining information, linear scoring modelsare unfortunately, plagued by the flat-maximum effect or the‘curse of insensitivity’. As Lovie & Lovie(1986)observe: ‘The predictive ability of linear models is insensitiveto large variations in the size of regression weights and tothe number of predictors.’ In essence, seemingly differentscoringmodels tend to produce indistinguishable predictive outcomes. Since its demonstration by Dawes & Corrigan (1974), observershave cast the flat maximum in a decidedly negative light. Incontrast, Lovie & Lovie (1986) present a provocatively contrarianview of the flat maximum‘s positive potential. In thissame vein, we examine the predictive power of a generic credit-scoringmodel versus individual empirically derived systems. If, asWainer (1976) noted in regard to the flat maximum, ’itdon‘t make no nevermind’, generic credit-scoringmodels could provide cheaper alternatives to individual empiricallyderived models. During the period 1984–8, a series of linear credit-scoringmodels were developed for ten Southeastern U.S. credit unions.For each credit union, stepwise multiple regression was employedto select a subset of explanatory variables to be used in adiscriminant analysis. A generic credit-scoring equation wasdeveloped from the resulting discriminant analyses using weightedaverage coefficients from five systems. The predictive powerof the generic model was compared to the predictive power ofholdout sample of the five remaining credit-scoring models. In all cases, the generic model's performance was very closeto that of the empirically derived models. Thus, our findingssupport Lovie & Lovie's (1986) challenge to the conventionalwisdom that the flat maximum casts a pall on the successfulmodelling of judgement processes. Indeed, the flat maximum impliesa positive role for simpler, and hence cheaper, generic models.Although further research is needed, it should be possible todevelop hybrid models with generic cores that perform as wellas empirically derived linear models.     

Jackknifed random weighting for Cox proportional hazards model

The Cox proportional hazards model is the most used statistical model in the analysis of survival time data.Recently,a random weighting method was proposed to approximate the distribution of the maximum partial likelihood estimate for the regression coefficient in the Cox model.This method was shown not as sensitive to heavy censoring as the bootstrap method in simulation studies but it may not be second-order accurate as was shown for the bootstrap approximation.In this paper,we propose an alternative random weighting method based on one-step linear jackknife pseudo values and prove the second accuracy of the proposed method.Monte Carlo simulations are also performed to evaluate the proposed method for fixed sample sizes.     

Efficient frontier cutoff policies in credit portfolios

Historically, account acquisition in scored retail credit and loan portfolios has focused on risk management in the sense of minimizing default losses. We believe that acquisition policies should focus on a broader set of business measures that explicitly recognize tradeoffs between conflicting objectives of losses, volume and profit. Typical business challenges are: ‘How do I maximize portfolio profit while keeping acceptance rate (volume, size) at acceptable levels?’ ‘How do I maximize profit without incurring default losses above a given level?’ ‘How do I minimize the risk of large loss exposures for a given market share?’ In this paper we are not concerned with which combination of objectives are appropriate, but rather focus on the cutoff policies that allow us to capture a number of different portfolio objectives. When there are conflicting objectives we show that optimal policies yield meaningful tradeoffs and efficient frontiers and that optimal shadow prices allow us to develop risk-adjusted tradeoffs between profit and market share. Some of the graphical solutions that we obtain are simple to derive and easy to understand without explicit mathematical formulations but even simple constraints may require formal use of non-linear programming techniques. We concentrate on models and insights that yield decision strategies and cutoff policies rather than the techniques for developing good predictors.     

A semiparametric Bernstein–von Mises theorem for Gaussian process priors

This paper is a contribution to the Bayesian theory of semiparametric estimation. We are interested in the so-called Bernstein–von Mises theorem, in a semiparametric framework where the unknown quantity is (θ, f), with θ the parameter of interest and f an infinite-dimensional nuisance parameter. Two theorems are established, one in the case with no loss of information and one in the information loss case with Gaussian process priors. The general theory is applied to three specific models: the estimation of the center of symmetry of a symmetric function in Gaussian white noise, a time-discrete functional data analysis model and Cox’s proportional hazards model. In all cases, the range of application of the theorems is investigated by using a family of Gaussian priors parametrized by a continuous parameter.