Modelling LGD for unsecured retail loans using Bayesian methods

Loss Given Default (LGD) is the loss borne by the bank when a customer defaults on a loan. LGD for unsecured retail loans is often found difficult to model. In the frequentist (non-Bayesian) two-step approach, two separate regression models are estimated independently, which can be considered potentially problematic when trying to combine them to make predictions about LGD. The result is a point estimate of LGD for each loan. Alternatively, LGD can be modelled using Bayesian methods. In the Bayesian framework, one can build a single, hierarchical model instead of two separate ones, which makes this a more coherent approach. In this paper, Bayesian methods as well as the frequentist approach are applied to the data on personal loans provided by a large UK bank. As expected, the posterior means of parameters that have been produced using Bayesian methods are very similar to the frequentist estimates. The most important advantage of the Bayesian model is that it generates an individual predictive distribution of LGD for each loan. Potential applications of such distributions include the downturn LGD and the stressed LGD under Basel II.     

Forecasting Loss Given Default models: impact of account characteristics and the macroeconomic state

On the basis of two data sets containing Loss Given Default (LGD) observations of home equity and corporate loans, we consider non-linear and non-parametric techniques to model and forecast LGD. These techniques include non-linear Support Vector Regression (SVR), a regression tree, a transformed linear model and a two-stage model combining a linear regression with SVR. We compare these models with an ordinary least squares linear regression. In addition, we incorporate several variants of 11 macroeconomic indicators to estimate the influence of the economic state on loan losses. The out-of-time set-up is complemented with an out-of-sample set-up to mitigate the limited number of credit crisis observations available in credit risk data sets. The two-stage/transformed model outperforms the other techniques when forecasting out-of-time for the home equity/corporate data set, while the non-parametric regression tree is the best performer when forecasting out-of-sample. The incorporation of macroeconomic variables significantly improves the prediction performance. The downturn impact ranges up to 5% depending on the data set and the macroeconomic conditions defining the downturn. These conclusions can help financial institutions when estimating LGD under the internal ratings-based approach of the Basel Accords in order to estimate the downturn LGD needed to calculate the capital requirements. Banks are also required as part of stress test exercises to assess the impact of stressed macroeconomic scenarios on their Profit and Loss (P&L) and banking book, which favours the accurate identification of relevant macroeconomic variables driving LGD evolutions.     

汽车消费、住房抵押贷款违约特性及对策

本通过对比住房抵押贷款和汽车消费贷款的违约特性，得出在利率由央行统一规定的条件下，对住房抵押贷款和汽车消费贷款，实行同样的首付款政策是不合理的，这导致目前汽车消费贷款的违约率居高不下。对于住房抵押贷款，银行可以适当降低首付款，来提高本银行住房抵押贷款在市场的竞争力，对于汽车消费贷款，可以通过采取提高汽车贷款首付款的措施，来降低违约率，控制抵押贷款风险。     

Quantile regression for modelling distributions of profit and loss

Quantile regression is applied in two retail credit risk assessment exercises exemplifying the power of the technique to account for the diverse distributions that arise in the financial service industry. The first application is to predict loss given default for secured loans, in particular retail mortgages. This is an asymmetric process since where the security (such as a property) value exceeds the loan balance the banks cannot retain the profit, whereas when the security does not cover the value of the defaulting loan then the bank realises a loss. In the light of this asymmetry it becomes apparent that estimating the low tail of the house value is much more relevant for estimating likely losses than estimates of the average value where in most cases no loss is realised. In our application quantile regression is used to estimate the distribution of property values realised on repossession that is then used to calculate loss given default estimates. An illustration is given for a mortgage portfolio from a European mortgage lender. A second application is to revenue modelling. While credit issuing organisations have access to large databases, they also build models to assess the likely effects of new strategies for which, by definition, there is no existing data. Certain strategies are aimed at increasing the revenue stream or decreasing the risk in specific market segments. Using a simple artificial revenue model, quantile regression is applied to elucidate the details of subsets of accounts, such as the least profitable, as predicted from their covariates. The application uses standard linear and kernel smoothed quantile regression.     

Credit scoring with macroeconomic variables using survival analysis

Survival analysis can be applied to build models for time to default on debt. In this paper, we report an application of survival analysis to model default on a large data set of credit card accounts. We explore the hypothesis that probability of default (PD) is affected by general conditions in the economy over time. These macroeconomic variables (MVs) cannot readily be included in logistic regression models. However, survival analysis provides a framework for their inclusion as time-varying covariates. Various MVs, such as interest rate and unemployment rate, are included in the analysis. We show that inclusion of these indicators improves model fit and affects PD yielding a modest improvement in predictions of default on an independent test set.     

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.     

Intensity models and transition probabilities for credit card loan delinquencies

We estimate the probability of delinquency and default for a sample of credit card loans using intensity models, via semi-parametric multiplicative hazard models with time-varying covariates. It is the first time these models, previously applied for the estimation of rating transitions, are used on retail loans. Four states are defined in this non-homogenous Markov chain: up-to-date, one month in arrears, two months in arrears, and default; where transitions between states are affected by individual characteristics of the debtor at application and their repayment behaviour since. These intensity estimations allow for insights into the factors that affect movements towards (and recovery from) delinquency, and into default (or not). Results indicate that different types of debtors behave differently while in different states. The probabilities estimated for each type of transition are then used to make out-of-sample predictions over a specified period of time.     

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.     

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.     

中国住房抵押贷款信用风险:理论分析与实证研究

住房抵押贷款为中国经济的持续增长增添了新的动力,随着规模扩大,其信用风险问题已经引起金融机构、政府部门及学者的关注.在分析中国房地产市场特点的基础上研究了适应中国住房抵押贷款违约的理论以及影响住房抵押贷款违约的因素,并通过采集大连市的数据进行了实证分析,首次运用实际数据来比选适应中国市场的理论模型.我们的研究发现:在中国住房抵押贷款市场上,贷款违约的还款能力理论较之于期权理论有着更好的适应性;利率、LTV、偿债比与户籍是影响住房抵押贷款违约的主要因素;也得出另外几个不同于理论假说的结论:家庭收入对借款人违约的影响力不明显,购买二手住房的借款人的违约概率要比新房高.     

P2P网贷违约人是否具有区域性特征——来自湖南省的例证

借助网络爬虫技术手段获取"人人贷"平台上借款人的各项信息,提取两个样本:分为全国随机样本和湖南省随机样本,构建二元Logit回归模型,分析其中对违约率有显著影响的变量.研究表明,负债收入比、借款期限、学历、房产、房贷、描述指数对违约行为有负向影响,而借款利率、车产、认证个数对借款者违约行为有正向影响.同时,通过对两个样本最终回归模型的比较,发现湖南省违约人特征与全国随机样本中体现的违约人特征基本一致,但其中较为特殊的是,在湖南拥有房产和车产不能作为网络借款人履约能力提升的标志.     

Modelling credit risk of portfolio of consumer loans

One of the issues that the Basel Accord highlighted was that, though techniques for estimating the probability of default and hence the credit risk of loans to individual consumers are well established, there were no models for the credit risk of portfolios of such loans. Motivated by the reduced form models for credit risk in corporate lending, we seek to exploit the obvious parallels between behavioural scores and the ratings ascribed to corporate bonds to build consumer-lending equivalents. We incorporate both consumer-specific ratings and macroeconomic factors in the framework of Cox Proportional Hazard models. Our results show that default intensities of consumers are significantly influenced by macro factors. Such models then can be used as the basis for simulation approaches to estimate the credit risk of portfolios of consumer loans.     

基于混合模型的利润驱动违约判别临界点研究

违约判别临界点是金融机构是否接受客户贷款申请的重要参考,合适的违约判别临界点对减少金融机构贷款损失实现稳健经营具有重要意义。本文研究的问题是如何保证计算客户违约概率的准确性,并找到利润最大化的违约判别临界点。本文的创新与特色:一是通过将多个不同类型的违约判别模型计算的客户违约概率进行加权平均,保证了计算客户违约概率的的整体准确性,避免了使用单一模型计算客户违约概率不准确的弊端;二是通过定义金融机构从贷款中获得利润的计算公式,以利润最大为目标,求解违约判别临界点,避免了现有计算临界点的方法如广义对称点估计和经验似然法等方法得到的临界点利润不是最大的弊端。研究发现:混合模型比单一模型的准确性高,AUC值显著提高;在人人贷数据集中本文的违约判别临界点下贷款利润远高于其他方法下临界点的利润。     

A generic framework for stochastic Loss-Given-Default

In this document a method is discussed to incorporate stochastic Loss-Given-Default (LGD) in factor models, i.e. structural models for credit risk. The general idea exhibited in this text is to introduce a common dependence of the LGD and the probability of default (PD) on a latent variable, representing the systemic risk. Though our theory can be applied to any arbitrary firm-value model and any underlying distribution for the LGD, provided its support is a compact subset of [0,1], special attention is given to the extension of the well-known cases of the Gaussian copula framework and the shifted Gamma one-factor model (a particular case of the generic one-factor Lévy model), and the LGD is modeled by a Beta distribution, in accordance with rating agency models and the Credit Metrics model.In order to introduce stochastic LGD, a monotonically decreasing relation is derived between the loss rate L, i.e. the loss as a percentage of the total exposure, and the standardized log-return R of the obligor’s asset value, which is assumed to be a function of one or more systematic and idiosyncratic risk factors. The property that the relation is decreasing guarantees that the LGD is negatively correlated to R and hence positively correlated to the default rate. From this relation, expressions are then derived for the cumulative distribution function (CDF) and the expected value of the loss rate and the LGD, conditionally on a realization of the systematic risk factor(s). It is important to remark that all our results are derived under the large homogeneous portfolio (LHP) assumption and that they are fully consistent with the IRB approach outlined by the Basel II Capital Accord.We will demonstrate the impact of incorporating stochastic LGD and using models based on skew and fat-tailed distributions in determining adequate capital requirements. Furthermore, we also skim the potential application of the proposed framework in a credit risk environment. It will turn out that both building blocks, i.e. stochastic LGD and fat-tailed distributions, separately, increase the projected loss and thus the required capital charge. Hence, the aggregation of a model based on a fat-tailed underlying distribution that accounts for stochastic LGD will lead to sound capital requirements.     

Modeling credit portfolio derivatives,including both a default and a prepayment feature

Apart from heteronomy exit events such as, for example credit default or death, several financial agreements allow policy holders to voluntarily terminate the contract. Examples include callable mortgages or life insurance contracts. For the contractual counterpart, the result is a cash‐flow uncertainty called prepayment risk. Despite the high relevance of this implicit option, only few portfolio models consider both a default and a cancellability feature. On a portfolio level, this is especially critical because empirical observations of the mortgage market suggest that prepayment risk is an important determinant for the pricing of mortgage‐backed securities. Furthermore, defaults and prepayments tend to occur in clusters, and there is evidence for a negative association between the two risks. This paper presents a realistic and tractable portfolio model that takes into account these observations. Technically, we rely on an Archimedean dependence structure. A suitable parameterization allows to fit the likelihood of default and prepayment clusters separately and accounts for the postulated negative interdependence. Moreover, this structure turns out to be tractable enough for real‐time evaluation of portfolio derivatives. As an application, the pricing of loan credit default swaps, an example of a portfolio derivative that includes a cancellability feature, is discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

Development and application of consumer credit scoring models using profit-based classification measures

This paper presents a new approach for consumer credit scoring, by tailoring a profit-based classification performance measure to credit risk modeling. This performance measure takes into account the expected profits and losses of credit granting and thereby better aligns the model developers’ objectives with those of the lending company. It is based on the Expected Maximum Profit (EMP) measure and is used to find a trade-off between the expected losses – driven by the exposure of the loan and the loss given default – and the operational income given by the loan. Additionally, one of the major advantages of using the proposed measure is that it permits to calculate the optimal cutoff value, which is necessary for model implementation. To test the proposed approach, we use a dataset of loans granted by a government institution, and benchmarked the accuracy and monetary gain of using EMP, accuracy, and the area under the ROC curve as measures for selecting model parameters, and for determining the respective cutoff values. The results show that our proposed profit-based classification measure outperforms the alternative approaches in terms of both accuracy and monetary value in the test set, and that it facilitates model deployment.     

Predicting bank loan recovery rates with a mixed continuous‐discrete model

To represent the high concentration of recovery rates at the boundaries, we propose to consider the recovery rate as a mixed random variable, obtained as the mixture of a Bernoulli random variable and a beta random variable. We suggest to estimate the mixture weights and the Bernoulli parameter by two logistic regression models. For the recovery rates belonging to the interval (0,1), we model, jointly, the mean and the dispersion by using two link functions, so we propose the joint beta regression model that accommodates skewness and heteroscedastic errors. This methodological proposal is applied to a comprehensive survey on loan recovery process of Italian banks. In the regression model, we include some macroeconomic variables because they are relevant to explain the recovery rate and allow to estimate it in downturn conditions, as Basel II requires. Copyright © 2012 John Wiley & Sons, Ltd.     

Solutions to specification errors in stress testing models

The regulatory and business need to expand the use of macroeconomic-scenario-based forecasting and stress testing in retail lending has led to a rapid expansion in the types and complexity of models being applied. As these models become more sophisticated and include lifecycle, credit quality, and macroeconomic effects, model specification errors become a common, but rarely identified feature of many of these models. This problem was discovered decades ago in demography with Age-Period-Cohort (APC) models, and we bring those insights to the retail lending context with a detailed discussion of the implications here. Although the APC literature proves that no universal, data-driven solution is possible, we propose a domain-specific solution that is appropriate to lending. This solution is demonstrated with an auto loan portfolio.     

基于区间数DEA的贷款定价模型研究

在贷款的买方市场或充分竞争的金融环境中,贷款利率不会由银行自己说了算,因此建立银企双方共同接受的贷款利率定价模型在现实中尤为重要。本文采用区间数的形式反映存款利息支出率、违约风险补偿率等定价指标的不确定性,以已结清贷款最小定价效率、最大定价效率组成的贷款定价效率区间为目标,以新贷款的贷款利率为决策变量,通过逆向求解区间数DEA模型反推出新贷款的贷款利率区间,建立了基于区间数DEA的贷款定价模型。本文的创新与特色一是以已结清贷款的存款利息支出率、目标利润率等指标为输入,以已结清贷款的贷款利率为输出,利用DEA模型求得已结清贷款的实际最小效率及最大效率。二是以银企双方均可接受的贷款定价效率区间为目标、以新贷款的存款利息支出率等用区间数形式表示的贷款成本为投入,反推出贷款利率的取值区间。三是通过区间数形式来反映违约风险补偿率、目标利润率等定价指标的不确定性,改变了现有研究将目标利润、贷款费用、违约损失等变量看作常数来定价的不合理现状。研究表明:存款利息支出率、费用支出率、违约风险补偿率及目标利润率均与贷款利率成正比。企业提高在贷款银行中的资金结算比率、存贷比率可以降低贷款利率。     

基于障碍期权的贷款保险定价研究

现有的贷款保险定价模型通常忽略了违约门槛和提前违约对贷款损失的影响。本文基于障碍期权中的向下敲入看跌期权,将这两个重要因素纳入到了新的贷款保险定价模型中。进一步,本文通过蒙特卡洛模拟的方法,给出了贷款保险敲入概率和敲入时间点的估计过程。此外,本文将新构建的贷款保险定价模型应用于实际中,并进行了实证分析。结果表明,违约门槛的上升会提高贷款保险的定价水平和敲入概率,并导致更早的敲入时间点。而银行降低对企业违约情况的观察频率会引起贷款保险的价值损失。