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.     

The structure of unrevealed Bads in Good/Bad risk scores

In calculating risk scores for making predictions and decisions about loan defaults, it is common practice to base assessments on a population of individuals whose loans have not yet attained a final status or trapped state of Good (G: paid in full) or Bad (B: default, bankrupt, written off, no response, etc). When active accounts are examined prior to end of loan term, we describe them as Contaminated Goods (CG) because they contain some Bads that default at a later time. In such cases, one can easily misestimate or misinterpret the eventual population odds and scores because the CG to B odds at any point in time is larger than G to B at the end of the loan. It is shown that if the risk score is a sufficient statistic and if the Information Odds score for Goods at the end-of-term is normal with variance σ² in a population of terminated loan accounts, then so also is the conditional score distribution for Bads; surprisingly, the theoretical means are ±0.5σ². When active accounts are contaminated by unrevealed Bads not yet classified as such, the conditional score distribution is a mixture of normal distributions with a variance larger than σ²; thus, variances of Active (CG) and Bad (B) accounts are unequal and the log of fitted odds versus score is convex, departing from the traditional assumption of a linear fit.     

The network structure and systemic risk in the global non-life insurance market

This paper contributes to the literature on systemic risk by assessing the systemic importance of insurers in the global non-life insurance market. First, we estimate the bilateral reinsurance claims matrix using the aggregate outstanding reinsurance data from ISIS and theoretically analyze the interconnectedness in the global reinsurance network using network indicators. The robustness of the estimated matrix is fully assured by sensitivity analysis. Second, we theoretically analyze the contagious defaults introducing the Eisenberg–Noe framework. Reinsurers play a dominant role in the reinsurance network and most of them are included in our data sample. The network analysis finds that some reinsurers with large centrality measures are central in the hierarchical structure of the network. The default analysis shows the occurrences of many stand-alone defaults and only one contagious default via the global reinsurance network after the global financial crisis. In addition, one stress test based on a hypothetical severe stress scenario predicts a few occurrences of contagious defaults in the future. It follows from these analyses that systemic risk via the global reinsurance network is relatively restricted in the global non-life insurance market. In conclusion, our methodology would help supervisory authorities develop an assessment approach for interconnectedness in the global reinsurance network and aid the implementation of insurer stress tests for default contagion.     

Adverse selection and non-take inference with coherent risk and response scoring

The authors offer a mathematical model for adverse selection by individual borrowers based on preferences for offers and the default (Bad) or non-default (Good) status of booked accounts. We define the condition for borrower risk and response when there is no adverse selection (NAS). This definition provides us with a direct comparison between the prior and posterior conditional probabilities of default by an individual borrower who Takes an offer; this allows us to obtain estimates of differential response rates for individual borrowers and the Good/Bad odds for Take, Non-Take and Accept sub-populations. Performance of different response-risk segments allows us to compare price-driven risk elasticity and price-driven response elasticity in the presence of Good or Bad adverse selections; a special case applies when the borrower's capacity to repay is not an issue. We offer limited experimental results for selected price-risk segments where action-based risk and response scores are used to estimate borrower preferences. The critical role of Non-Take inference is described.     

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 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.     

Default risk analysis via a discrete‐time cure rate model

Cure models represent an appealing tool when analyzing default time data where two groups of companies are supposed to coexist: those which could eventually experience a default (uncured) and those which could not develop an endpoint (cured). One of their most interesting properties is the possibility to distinguish among covariates exerting their influence on the probability of belonging to the populations’ uncured fraction, from those affecting the default time distribution. This feature allows a separate analysis of the two dimensions of the default risk: whether the default can occur and when it will occur, given that it can occur. Basing our analysis on a large sample of Italian firms, the probability of being uncured is here estimated with a binary logit regression, whereas a discrete time version of a Cox's proportional hazards approach is used to model the time distribution of defaults. The extension of the cure model as a forecasting framework is then accomplished by replacing the discrete time baseline function with an appropriate time‐varying system level covariate, able to capture the underlying macroeconomic cycle. We propose a holdout sample procedure to test the classification power of the cure model. When compared with a single‐period logit regression and a standard duration analysis approach, the cure model has proven to be more reliable in terms of the overall predictive performance. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Samsung card lending model

Samsung Card Lending Model (SCLM) analyzes cash flow in individual accounts and measures the level of company-wide risk. Serving as a risk and portfolio management model in the consumer lending business, the main features of SCLM are as follows. Default ratios such as intrinsic balance default probability and annual default ratio are computed using the past, present, and future cash flows of accounts. The provision is shown as the total sum of write-offs. The size of capital required is determined by default probability distribution. The price for new accounts is quoted based on cash flow simulations reflecting future business environments. SCLM has shown good performance in Samsung card consumer lending business since the Korean credit card crisis of 2003.     

Counterparty risk valuation on credit-linked notes under a Markov Chain framework

A credit-linked note(CLN) is a note paying an enhanced coupon to investors for bearing the credit risk of a reference entity. In this paper, we study the counterparty risk on CLNs under a Markov chain framework, and introduce a Markov copula model to describe joint defaults between the reference entity underlying the CLN and CLN issuer. Assuming that the respective default intensities are directly and inversely proportional to the interest rate, which follows a CIR process, we obtain the explicit formulae for CLN values through a PDE approach.Finally, credit valuation adjustment(CVA) formula is derived to price counterparty credit risk.     

利用传染模型对含有对手风险的信用证券定价(英文)

本文利用传染模型研究了可违约债券和含有对手风险的信用违约互换的定价。我们在约化模型中引入具有违约相关性的传染模型,该模型假设违约过程的强度依赖于由随机微分方程驱动的随机利率过程和交易对手的违约过程.本文模型可视为Jarrow和Yu(2001)及Hao和Ye(2011)中模型的推广.进一步地,我们利用随机指数的性质导出了可违约债券和含有对手风险的信用违约互换的定价公式并进行了数值分析.     

Credit risk model with contagious default dependencies affected by macro-economic condition

We consider a credit risk model with two industrial sectors, where defaults of corporations would be influenced by two factors. The first factor represents the macro economic condition which would affect the default intensities of the two industrial sectors differently. The second factor reflects the influences of the past defaults of corporations against other active corporations, where such influences would affect the two industrial sectors differently. A two-layer Markov chain model is developed, where the macro economic condition is described as a birth-death process, while another Markov chain represents the stochastic characteristics of defaults with default intensities dependent on the state of the birth-death process and the number of defaults in two sectors. Although the state space of the two-layer Markov chain is huge, the fundamental absorbing process with a reasonable state space size could capture the first passage time structure of the two-layer Markov chain, thereby enabling one to evaluate the joint probability of the number of defaults in two sectors via the uniformization procedure of Keilson. This in turn enables one to value a variety of derivatives defined on the underlying credit portfolios. In this paper, we focus on a financial product called CDO, and a related option.     

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.     

Copula方法与相依违约研究

目前信用风险研究的重点已经从单笔债务的违约概率研究转移到多笔债务的相依违约(Dependent Defaults)研究。Copula方法是研究相依违约的重要方法。这种方法是最近几年才被应用到信用领域研究中的一种新方法。本结合代表性献对Copula方法在相依违约研究中的应用进行了探讨。探讨的内容包括Copula方法被应用于相依违约研究的原因、该方法对于相依违约建模理论的改进以及在实证应用中使用Copula方法应该注意的问题。     

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.     

债券违约是否存在区域传染？——基于中国债券市场融资成本视角

2018年以来中国债券市场违约规模攀升,累计违约金额超2900亿元。债券违约后的负面影响受到投资者、发行人乃至监管部门关注。本文以北京、上海、辽宁等八个辖区为例,选取2016~2019年债券违约及债券发行数据,通过构建违约事件对债券发行价格影响因素模型,分析了债券违约的区域传染效应。研究发现:债券违约引发的信用风险存在区域传染性,主要体现为债券发行前若发行人所属辖区存在违约事件将推升债券融资成本;区域内的传染效应与违约时间距离负相关,当时间距离增长时,传染效应变弱,甚至消失;债券违约风险对不同性质企业的传染效应不同,民营企业受影响尤为显著。     

Tail behaviour of credit loss distributions for general latent factor models

Using a limiting approach to portfolio credit risk, we obtain analytic expressions for the tail behavior of credit losses. To capture the co‐movements in defaults over time, we assume that defaults are triggered by a general, possibly non‐linear, factor model involving both systematic and idiosyncratic risk factors. The model encompasses default mechanisms in popular models of portfolio credit risk, such as CreditMetrics and CreditRisk⁺. We show how the tail characteristics of portfolio credit losses depend directly upon the factor model's functional form and the tail properties of the model's risk factors. In many cases the credit loss distribution has a polynomial (rather than exponential) tail. This feature is robust to changes in tail characteristics of the underlying risk factors. Finally, we show that the interaction between portfolio quality and credit loss tail behavior is strikingly different between the CreditMetrics and CreditRisk⁺ approach to modeling portfolio credit risk.     

Random effects logistic regression model for default prediction of technology credit guarantee fund

Korean government has been funding the small and medium enterprises (SME) with superior technology based on scorecard. However high default rate of funded SMEs has been reported. In order to effectively manage such governmental fund, it is important to develop accurate scoring model for SMEs. In this paper, we provide a random effects logistic regression model to predict the default of funded SMEs based on both financial and non-financial factors. Advantage of such a random effects model lies in the ability of accommodating not only the individual characteristics of each SME but also the uncertainty that cannot be explained by such individual factors. It is expected that our study can contribute to effective management of government funds by proposing the prediction models for defaults of funded SMEs.     

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.     

A dynamic binomial expansion technique for credit risk measurement: a Bayesian filtering approach

Credit risk measurement and management are important and current issues in the modern finance world from both the theoretical and practical perspectives. There are two major schools of thought for credit risk analysis, namely the structural models based on the asset value model originally proposed by Merton and the intensity‐based reduced form models. One of the popular credit risk models used in practice is the Binomial Expansion Technique (BET) introduced by Moody's. However, its one‐period static nature and the independence assumption for credit entities' defaults are two shortcomings for the use of BET in practical situations. Davis and Lo provided elegant ways to ease the two shortcomings of BET with their default infection and dynamic continuous‐time intensity‐based approaches. This paper first proposes a discrete‐time dynamic extension to the BET in order to incorporate the time‐dependent and time‐varying behaviour of default probabilities for measuring the risk of a credit risky portfolio. In reality, the ‘true’ default probabilities are unobservable to credit analysts and traders. Here, the uncertainties of ‘true’ default probabilities are incorporated in the context of a dynamic Bayesian paradigm. Numerical studies of the proposed model are provided.