Bayesian inference for generalized linear mixed models of portfolio credit risk

The aims of this paper are threefold. First, we highlight the usefulness of generalized linear mixed models (GLMMs) in the modelling of portfolio credit default risk. The GLMM-setting allows for a flexible specification of the systematic portfolio risk in terms of observed fixed effects and unobserved random effects, in order to explain the phenomena of default dependence and time-inhomogeneity in historical default data. Second, we show that computational Bayesian techniques such as the Gibbs sampler can be successfully applied to fit models with serially correlated random effects, which are special instances of state space models. Third, we provide an empirical study using Standard and Poor's data on U.S. firms. A model incorporating rating category and sector effects, and a macroeconomic proxy variable for state-of-the-economy suggests the presence of a residual, cyclical, latent component in the systematic risk.     

Discrete credit barrier models

The model introduced in this article is designed to provide a consistent representation for both the real-world and pricing measures for the credit process. We find that good agreement with historical and market data can be achieved across all credit ratings simultaneously. The model is characterized by an underlying stochastic process that takes on values on a discrete lattice and represents credit quality. Rating transitions are associated with barrier crossings and default events are associated with an absorbing state. The stochastic process has state-dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift is estimated to consistently match the average spread curves corresponding to all the various ratings.     

Systemic risk contributions: A credit portfolio approach

We put forward a framework for measuring systemic risk and attributing it to individual banks. Systemic risk is coherently measured as the expected loss to depositors and investors when a systemic event occurs. The risk contributions are calculated so as to ensure a full risk allocation among institutions. Applying our methodology to a panel of 54–86 of the world’s major commercial banks for a 13-year time span with monthly frequency not only allows us to closely match the list of G-SIBs; we can also use individual risk contributions to compute bank-specific surcharges: systemic capital charges as well as countercyclical buffers. We therefore address both dimensions of systemic risk – cross-sectional and time-series – in a single integrated approach. As the analysis of risk drivers confirms, the main focus of macroprudential supervision should be on a solid capital base throughout the financial cycle and de-correlation of banks’ asset values.     

Granularity adjustment for mark-to-market credit risk models

The impact of undiversified idiosyncratic risk on value-at-risk and expected shortfall can be approximated analytically via a methodology known as granularity adjustment (GA). In principle, the GA methodology can be applied to any risk-factor model of portfolio risk. Thus far, however, analytical results have been derived only for simple models of actuarial loss, i.e., credit loss due to default. We demonstrate that the GA is entirely tractable for single-factor versions of a large class of models that includes all the commonly used mark-to-market approaches. Our approach covers both finite ratings-based models and models with a continuum of obligor states. We apply our methodology to CreditMetrics and KMV Portfolio Manager, as these are benchmark models for the finite and continuous classes, respectively. Comparative statics of the GA reveal striking and counterintuitive patterns. We explain these relationships with a stylized model of portfolio risk.     

A semiparametric model for the systematic factors of portfolio credit risk premia

The aim of this paper is to investigate the empirical relationship between daily fluctuations in the risk premium for holding a large diversified credit portfolio, which we approximate by a benchmark credit index, and some tradeable market factors which capture systematic risk. The analysis is based on an adaptive nonparametric modelling approach which allows for the data-driven estimation of the nonlinear dynamic relationship between portfolio credit risk premia and their hypothetical components. Our main finding is that the empirical weights of the systematic factors display sudden jumps during market crises and a less intense time-dependent behaviour during normal market conditions. In addition, we find that during market crises the directions of the empirical relationships are often inconsistent with ordinary economic intuition, as they are influenced by the specific circumstances of financial markets distress.     

Measuring portfolio credit risk correctly: Why parameter uncertainty matters

Why should risk management systems account for parameter uncertainty? In addressing this question, the paper lets an investor in a credit portfolio face non-diversifiable uncertainty about two risk parameters – probability of default and asset-return correlation – and calibrates this uncertainty to a lower bound on estimation noise. In this context, a Bayesian inference procedure is essential for deriving and analyzing the main result, i.e. that parameter uncertainty raises substantially the tail risk perceived by the investor. Since a measure of tail risk that incorporates parameter uncertainty is computationally demanding, the paper also derives a closed-form approximation to such a measure.     

A simple continuous measure of credit risk

This paper introduces a simple parameterization for the risk-neutral default probability distributions for risky firms that are easily computed from quoted bond prices. The corresponding expected times to default have a particularly simple form and are proposed as a measure for credit risk. Being continuous in nature, times to default provide a much finer measure of risk than those provided by ratings agencies. Comparison with the ratings provided by Moody's and the distance to default measures calculated using the Merton [Merton, R. (1974). On the pricing of corporate debt: the risk structure of interest rates. Journal of Finance, 2(2), 449-470] model shows that the highest rank correlation is found between the proposed time to default measure and Moody's ratings.     

Contagion effect on bond portfolio risk measures in a hybrid credit risk model

This paper illustrates how modelling the contagion effect among assets of a given bond portfolio changes the risk perception associated to it. This empirical work is developed in a hybrid credit risk framework that incorporates recovery rate risk. Dependence structures among firms and between external shocks affecting firms together are considered. The presence of correlations among firm leverage ratios and the interrelation between default probabilities and recovery rates produces clusters of defaults with low recovery rates. This has a major impact on standard risk measures such as Value-at-Risk and conditional tail expectation. Consequently, an appropriate measurement of the contagion has a tremendous effect on the capital requirement of many financial institutions.     

Laying off credit risk: Loan sales versus credit default swaps

How do markets for debt cash flow rights, with and without accompanying control rights, affect the efficiency of lending? A bank makes a loan, learns if it needs monitoring, and then decides whether to lay off credit risk. The bank can transfer credit risk by either selling the loan or buying a credit default swap (CDS). With a CDS, the originating bank retains the loan's control rights; with loan sales, control rights pass to the loan buyer. Credit risk transfer leads to excessive monitoring of riskier credits and insufficient monitoring of safer credits. Increases in banks' cost of equity capital exacerbate these effects. For riskier credits, loan sales typically dominate CDS but not for safer credits. Once repeated lending and consequent reputation concerns are modeled, although CDSs remain dominated by loan sales for riskier credits, for safer credits they can dominate loan sales, supporting better monitoring (albeit to a limited extent) while allowing efficient risk sharing. Restrictions on the bank's ability to sell the loan expand the range in which CDSs are used and monitoring is too low.     

采用信用风险模型计提监管资本:问题和对策

实施基于商业银行内部信用风险模型的资本监管制度对监管当局和商业银行都提出了严峻的挑战。信用风险计量模型不仅在理论上应经得起推敲,其计量结果在实证上需经得起检验,并且在不同银行间应具有可比性。本文为美联储的五位高级监管人员(Beverly J.Hirtle,Mark Levonian,Marc Saidenberg,Stefan Walter,David Wright)发表在《纽约联储经济政策评论》2001年3月号上的研究报告。该报告对采用风险计量模型计提信用风险资本所涉及的许多重大问题进行了深入讨论,揭示了内部评级法的技术原理,业界关于内部评级法的许多争论很大程度上都围绕这些问题展开,部分问题在新资本协议最终稿以及巴塞尔委员会发表的监管文件中可以找到答案,如时间跨度、使用测试、贷款损失准备处理、报告频率等;但有些问题仍未解决,讨论还在继续,如计量模型的具体形式、模型验证技术。本报告对于读者深入理解内部评级技术和监管要求具有很强指导作用。     

Bank credit risk and structural credit models: Agency and information asymmetry perspectives

This work investigates the effects of agency and information asymmetry issues embedded in structural form credit models on bank credit risk evaluation, using American bank data from 2001 to 2005. Findings show that both the agency problem and information asymmetry significantly cause deviations in the credit risk evaluation of structural form models from agency ratings. Five independent factors explain a deviation of 42.6–78.3% and should be incorporated into future credit risk modeling. Additionally, both the effects of information asymmetry and debt-equity agency positively relate to the deviation while that of management-equity agency relates to it negatively.     

Sensitivity estimates for portfolio credit derivatives using Monte Carlo

Portfolio credit derivatives are contracts that are tied to an underlying portfolio of defaultable reference assets and have payoffs that depend on the default times of these assets. The hedging of credit derivatives involves the calculation of the sensitivity of the contract value with respect to changes in the credit spreads of the underlying assets, or, more generally, with respect to parameters of the default-time distributions. We derive and analyze Monte Carlo estimators of these sensitivities. The payoff of a credit derivative is often discontinuous in the underlying default times, and this complicates the accurate estimation of sensitivities. Discontinuities introduced by changes in one default time can be smoothed by taking conditional expectations given all other default times. We use this to derive estimators and to give conditions under which they are unbiased. We also give conditions under which an alternative likelihood ratio method estimator is unbiased. We illustrate the application and verification of these conditions and estimators in the particular case of the multifactor Gaussian copula model, but the methods are more generally applicable.      

中国金融市场信用风险模型研究与应用

文章运用中国金融市场和原银行信贷登记系统数据及人民银行组织的信用评级等数据资源,在金融工程理论和技术方法创新的基础上,借鉴国际信用风险模型中违约模式代表——KMV模型原理,实证建立由判别函数和违约强度共同构成的中国金融市场违约预警模型;借鉴国际信用风险模型中盯市模式代表——CreditMetrics模型原理,使用蒙特卡罗模拟方法实证建立中国金融市场信用组合计量模型;探索这两类模型在中国信贷市场、外汇市场、货币市场和债券市场风险管理实务中的应用;并在此基础上提出了政策性建议。     

From insurance risk to credit portfolio management: a new approach to pricing CDOs

We present a new approach for pricing collateralized debt obligations (CDOs) which takes into account the issue of the market incompleteness. In particular, we develop a suitable extension of the actuarial framework proposed by Bayraktar et al. [Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities. J. Econ. Dyn. Control, 2009, 33, 676–691], Milevsky et al. [Financial valuation of mortality risk via the instantaneous Sharpe-ratio: Applications to pricing pure endowments. Working Paper, 2007. Available at: http://arxiv.org/abs/0705.1302], Young [Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio: Theorems and proofs. Technical Report, 2007. Available at: http://arxiv.org/abs/0705.1297] and Young [Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio. Insurance: Math. Econ., 2008, 42, 691–703], which is based on the so-called instantaneous Sharpe ratio. Such a procedure allows us to incorporate the attitude of investors towards risk in a direct and rational way and, in addition, is also suitable for dealing with the often illiquid CDO market. Numerical experiments are presented which reveal that the market incompleteness can have a strong effect on the pricing of CDOs, and allows us to explain the high bid-ask spreads that are frequently observed in the markets.     

Nonparametric correlation models for portfolio allocation

This article proposes time-varying nonparametric and semiparametric estimators of the conditional cross-correlation matrix in the context of portfolio allocation. Simulations results show that the nonparametric and semiparametric models are best in DGPs with substantial variability or structural breaks in correlations. Only when correlations are constant does the parametric DCC model deliver the best outcome. The methodologies are illustrated by evaluating two interesting portfolios. The first portfolio consists of the equity sector SPDRs and the S&P 500, while the second one contains major currencies. Results show the nonparametric model generally dominates the others when evaluating in-sample. However, the semiparametric model is best for out-of-sample analysis.     

Interacting particle systems for the computation of rare credit portfolio losses

In this paper, we introduce the use of interacting particle systems in the computation of probabilities of simultaneous defaults in large credit portfolios. The method can be applied to compute small historical as well as risk-neutral probabilities. It only requires that the model be based on a background Markov chain for which a simulation algorithm is available. We use the strategy developed by Del Moral and Garnier in (Ann. Appl. Probab. 15:2496–2534, 2005) for the estimation of random walk rare events probabilities. For the purpose of illustration, we consider a discrete-time version of a first passage model for default. We use a structural model with stochastic volatility, and we demonstrate the efficiency of our method in situations where importance sampling is not possible or numerically unstable.      

个人信用评分有力推进信贷风险管理转型

随着我国全面建设小康社会进程的加速,社会财富不断增加,个人客户对银行的个人业务提出了更多需求,进一步拉动了个人金融服务的增长.由于个人业务具有笔数多、单笔金额小,数据丰富的特点,所以要对其采用精细化、智能化、自动化的管理模式.然而长期以来,我国商业银行个人业务的经营管理方式一直比较粗放.中国人民银行最新研发的全国个人信用评分,可以对个人客户在未来一定时期内的违约概率进行评估和预测.它可以有效防范风险,为商业银行实现个人信贷风险管理转型提供先进的风险管理技术支持.     

Copula-based factor model for credit risk analysis

A standard quantitative method to assess credit risk employs a factor model based on joint multivariate normal distribution properties. By extending the one-factor Gaussian copula model to produce a more accurate default forecast, this paper proposes the incorporation of a state-dependent recovery rate into the conditional factor loading and to model them sharing a unique common factor. The common factor governs the default rate and recovery rate simultaneously, implicitly creating their association. In accordance with Basel III, this paper shows that the tendency toward default during a hectic period is governed more by systematic risk than by idiosyncratic risk. Among those considered, the model with random factor loading and a state-dependent recovery rate is shown to be superior in terms of default prediction.     

Checking for asymmetric default dependence in a credit card portfolio: A copula approach

Traditional credit risk models adopt the linear correlation as a measure of dependence and assume that credit losses are normally-distributed. However some studies have shown that credit losses are seldom normal and the linear correlation does not give accurate assessment for asymmetric data. Therefore it is possible that many credit models tend to misestimate the probability of joint extreme defaults.This paper employs Copula Theory to model the dependence across default rates in a credit card portfolio of a large UK bank and to estimate the likelihood of joint high default rates. Ten copula families are used as candidates to represent the dependence structure. The empirical analysis shows that, when compared to traditional models, estimations based on asymmetric copulas usually yield results closer to the ratio of simultaneous extreme losses observed in the credit card portfolio.Copulas have been applied to evaluate the dependence among corporate debts but this research is the first paper to give evidence of the outperformance of copula estimations in portfolios of consumer loans. Moreover we test some families of copulas that are not typically considered in credit risk studies and find out that three of them are suitable for representing dependence across credit card defaults.