The loss given default of a low-default portfolio with weak contagion

In this paper we study the loss given default (LGD) of a low default portfolio (LDP), assuming that there is weak credit contagion among the obligors. We characterize the credit contagion by a Sarmanov dependence structure of the risk factors that drive the obligors’ default, where the risk factors are assumed to be heavy tailed. From a new perspective of asymptotic analysis, we derive a limiting distribution for the LGD. As a consequence, an approximation for the entire distribution, in contrast to just the tail behavior, of the LGD is obtained. We show numerical examples to demonstrate the limiting distribution. We also discuss possible applications of the limiting distribution to the calculation of moments and the Value at Risk (VaR) of the LGD.     

Confidence sets and confidence bands for a beta distribution with applications to credit risk management

Incorporating statistical multiple comparisons techniques with credit risk measurement, a new methodology is proposed to construct exact confidence sets and exact confidence bands for a beta distribution. This involves simultaneous inference on the two parameters of the beta distribution, based upon the inversion of Kolmogorov tests. Some monotonicity properties of the distribution function of the beta distribution are established which enable the derivation of an efficient algorithm for the implementation of the procedure. The methodology has important applications to financial risk management. Specifically, the analysis of loss given default (LGD) data are often modeled with a beta distribution. This new approach properly addresses model risk caused by inadequate sample sizes of LGD data, and can be used in conjunction with the standard recommendations provided by regulators to provide enhanced and more informative analyses.     

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.     

The economy and loss given default: evidence from two UK retail lending data sets

Loss given default (LGD) models predict losses as a proportion of the outstanding loan, in the event a debtor goes into default. The literature on corporate sector LGD models suggests LGD is correlated to the economy and so changes in the economy could translate into different predictions of losses. In this work, the role of macroeconomic variables in loan-level retail LGD models is examined by testing the inclusion of macroeconomic variables in two different retail LGD models: a two-stage model for a residential mortgage loans data set and an ordinary least squares model for an unsecured personal loans data set. To improve loan-level predictions of LGD, indicators relating to the macroeconomy are considered with mixed results: the selected macroeconomic variable seemed able to improve the predictive performance of mortgage loan LGD estimates, but not for personal loan LGD. For mortgage loan LGD, interest rate was most beneficial but only predicted better during downturn periods, underestimating LGD during non-downturn periods. For personal loan LGD, only net lending growth is statistically significant but including this variable did not bring any improvement to R².     

Modelling Credit Risk in the Jump Threshold Framework

ABSTRACT

The jump threshold framework for credit risk modelling developed by Garreau and Kercheval enjoys the advantages of both structural- and reduced-form models. In their article, the focus is on multidimensional default dependence, under the assumptions that stock prices follow an exponential Lévy process (i.i.d. log returns) and that interest rates and stock volatility are constant. Explicit formulas for default time distributions and basket credit default swap (CDS) prices are obtained when the default threshold is deterministic, but only in terms of expectations when the default threshold is stochastic. In this article, we restrict attention to the one-dimensional, single-name case in order to obtain explicit closed-form solutions for the default time distribution when the default threshold, interest rate and volatility are all stochastic. When the interest rate and volatility processes are affine diffusions and the stochastic default threshold is properly chosen, we provide explicit formulas for the default time distribution, prices of defaultable bonds and CDS premia. The main idea is to make use of the Duffie–Pan–Singleton method of evaluating expectations of exponential integrals of affine diffusions.     

Stochastic models for risk estimation in volatile markets: a survey

Portfolio risk estimation in volatile markets requires employing fat-tailed models for financial returns combined with copula functions to capture asymmetries in dependence and an appropriate downside risk measure. In this survey, we discuss how these three essential components can be combined together in a Monte Carlo based framework for risk estimation and risk capital allocation with the average value-at-risk measure (AVaR). AVaR is the average loss provided that the loss is larger than a predefined value-at-risk level. We consider in some detail the AVaR calculation and estimation and investigate the stochastic stability.     

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.     

An incentive-compatible solution for trade credit term incorporating default risk

This paper studies the optimal trade credit term decision in an extended economic ordering quantity (EOQ) framework that incorporates a default risk component. A principal-agent bilevel programming model with costs minimization objectives is set up to derive the incentive-compatible credit term. The supplier determines the credit term as the leader in the first level programming, by balancing her/his financing capacity with the retailer’s default risk, order behavior and cost shifting. At the second level, the retailer makes decisions on ordering and payment time by reacting on the term offered by the supplier. A first order condition solution procedure is derived for the bilevel programming when credit term is confined within the practically feasible interval. Two key results are obtained – the condition to derive incentive-compatible credit term, and an equation system to derive threshold default risk criterion filtering retailers suitable for credit granting. Numerical experiments show that the capital cost of the supplier is the most important factor determining the credit term. Default risk acts like a filtering criterion for selecting retailers suitable for credit granting. Empirical evidence supporting our theoretical considerations is obtained by estimating three panel econometric models, using a dataset from China’s listed companies.     

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.     

基于双因子CIR强度式定价的信用债券投资组合优化

信用风险和利率风险是相互关联影响的。资产组合优化不能将这两种风险单独考虑或简单的相加,应该进行整体的风险控制,不然会造成投资风险的低估。本文的主要工作:一是在强度式定价模型的框架下,分别利用CIR随机利率模型刻画利率风险因素“无风险利率”和信用风险因素“违约强度”的随机动态变化,衡量在两类风险共同影响下信用债券的市场价值,从而构建CRRA型投资效用函数。以CRRA型投资效用函数最大化作为目标函数,同时控制利率和信用两类风险。弥补了现有研究中仅单独考虑信用风险或利率风险、无法对两种风险进行整体控制的弊端。二是将无风险利率作为影响违约强度的一个因子,利用“无风险利率因子”和“纯信用因子”的双因子CIR模型拟合违约强度,考虑了市场利率变化对于债券违约强度的影响,反映两种风险的相关性。使得投资组合模型中既同时考虑了信用风险和利率风险、又考虑了两种风险的交互影响。避免在优化资产组合时忽略两种风险间相关性、可能造成风险低估的问题。     

Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing

In a supplier-retailer-buyer supply chain, the supplier frequently offers the retailer a trade credit of S periods, and the retailer in turn provides a trade credit of R periods to her/his buyer to stimulate sales and reduce inventory. From the seller’s perspective, granting trade credit increases sales and revenue but also increases opportunity cost (i.e., the capital opportunity loss during credit period) and default risk (i.e., the percentage that the buyer will not be able to pay off her/his debt obligations). Hence, how to determine credit period is increasingly recognized as an important strategy to increase seller’s profitability. Also, many products such as fruits, vegetables, high-tech products, pharmaceuticals, and volatile liquids not only deteriorate continuously due to evaporation, obsolescence and spoilage but also have their expiration dates. However, only a few researchers take the expiration date of a deteriorating item into consideration. This paper proposes an economic order quantity model for the retailer where: (a) the supplier provides an up-stream trade credit and the retailer also offers a down-stream trade credit, (b) the retailer’s down-stream trade credit to the buyer not only increases sales and revenue but also opportunity cost and default risk, and (c) deteriorating items not only deteriorate continuously but also have their expiration dates. We then show that the retailer’s optimal credit period and cycle time not only exist but also are unique. Furthermore, we discuss several special cases including for non-deteriorating items. Finally, we run some numerical examples to illustrate the problem and provide managerial insights.     

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.     

A limit distribution of credit portfolio losses with low default probabilities

This paper employs a multivariate extreme value theory (EVT) approach to study the limit distribution of the loss of a general credit portfolio with low default probabilities. A latent variable model is employed to quantify the credit portfolio loss, where both heavy tails and tail dependence of the latent variables are realized via a multivariate regular variation (MRV) structure. An approximation formula to implement our main result numerically is obtained. Intensive simulation experiments are conducted, showing that this approximation formula is accurate for relatively small default probabilities, and that our approach is superior to a copula-based approach in reducing model risk.     

一类杠杆公司的破产概率:回望期权定价方法

利用结构化方法构造了杠杆公司的金融资产组合,由于公司破产的不可逆性和不确定性,可以把公司破产理解为公司所发行的债券发生违约.通过求解回望期权所满足的抛物型随机偏微分方程,推导出了混合分数跳-扩散模型下杠杆公司的股票定价公式,给出了杠杆公司在财务出现危机时股东通过资本注入来弥补经营损失和清偿债务而没有导致公司破产的概率,...     

Pricing catastrophe options with counterparty credit risk in a reduced form model

In this paper, we study the price of catastrophe options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model. We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.     

Modelling dependence structure with Archimedean copulas and applications to the iTraxx CDS index

In this paper we model the dependence structure between credit default swap (CDS) and jump risk using Archimedean copulas. The paper models and estimates the different relationships that can exist in different ranges of behaviour. It studies the bivariate distributions of CDS index spreads and the kurtosis of equity return distribution. To take into account nonlinear relationships and different structures of dependency, we employ three Archimedean copula functions: Gumbel, Clayton, and Frank. We adopt nonparametric estimation of copula parameters and we find an extreme co-movement of CDS and stock market conditions. In addition, tail dependence indicates the extreme co-movements and the potential for a simultaneous large loss in stock markets and a significant default risk. Ignoring the tail dependence would lead to underestimation of the default risk premium.     

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.     

Modeling the dynamics of interest rate volatility with skewed fat-tailed distributions

This paper proposes generalized parametric models of the short-term interest rate that nest one-factor CEV and discrete time GARCH models. The paper estimates the generalized and nested models with skewed fat-tailed distributions to determine the correct specification of the conditional distribution of interest rates. The results indicate that the discrete time models that incorporate the level and GARCH effects into the diffusion function and that accommodate the tail-thickness of the interest rate distribution perform much better than the CEV model in forecasting the future volatility of interest rates. The results also show that the significance of nonlinearity in the drift function relies crucially on the specification of the volatility function.     

具有共同冲击相依性的跳扩散金融市场中有着延迟和违约风险的鲁棒最优再保险和投资策略

在本文中,我们考虑跳扩散模型下具有延迟和违约风险的鲁棒最优再保险和投资问题,保险人可以投资无风险资产,可违约的债券和两个风险资产,其中两个风险资产遵循跳跃扩散模型且受到同种因素带来共同影响而相互关联.假设允许保险人购买比例再保险,特别地再保险保费利用均值方差保费原则来计算.在考虑与绩效相关的资本流入/流出下,保险公司的财富过程通过随机微分延迟方程建模.保险公司的目标是最大程度地发挥终端财富和平均绩效财富组合的预期指数效用,以分别研究违约前和违约后的情况.此外,推导了最优策略的闭式表达式和相应的价值函数.最后通过数值算例和敏感性分析,表明了各种参数对最优策略的影响.另外对于模糊厌恶投资者,忽视模型模糊性风险会带来显著的效用损失.     

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.