Modeling the evolution of implied CDO correlations

CDO tranche spreads (and prices of related portfolio-credit derivatives) depend on the market’s perception of the future loss distribution of the underlying credit portfolio. Applying Sklar’s seminal decomposition to the distribution of the vector of default times, the portfolio-loss distribution derived thereof is specified through individual default probabilities and the dependence among obligors’ default times. Moreover, the loss severity, specified via obligors’ recovery rates, is an additional determinant. Several (specifically univariate) credit derivatives are primarily driven by individual default probabilities, allowing investments in (or hedging against) default risk. However, there is no derivative that allows separately trading (or hedging) default correlations; all products exposed to correlation risk are contemporaneously also exposed to default risk. Moreover, the abstract notion of dependence among the names in a credit portfolio is not directly observable from traded assets. Inverting the classical Vasicek/Gauss copula model for the correlation parameter allows constructing time series of implied (compound and base) correlations. Based on such time series, it is possible to identify observable variables that describe implied correlations in terms of a regression model. This provides an economic model of the time evolution of the market’s view of the dependence structure. Different regression models are developed and investigated for the European CDO market. Applications and extensions to other markets are discussed.     

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.      

基于违约强度的固定收益证券风险敏感性分析与压力测试

固定收益证券存在单券投资和组合持仓规模较大的特殊性,势必面临着一定的市场风险、信用风险和流动性风险。本文通过实践方面的探索和尝试,改变了原有仅用收益率曲线平移的敏感性分析和压力测试方法,提出了利用基于违约强度的模型结合Vasicek利率模型的方法,以中债估值数据为例,对我国固定收益证券组合的市场(利率)风险和信用风险的敏感性分析和压力测试进行了检验。研究发现,在精确校准估值模型的前提下,均衡长期利率水平和均衡长期违约率越高,债券价格越低,均值回复速度越大和波动率越小,债券价格越低。     

Derivative Pricing with Liquidity Risk: Theory and Evidence from the Credit Default Swap Market

We derive an equilibrium asset pricing model incorporating liquidity risk, derivatives, and short‐selling due to hedging of nontraded risk. We show that illiquid assets can have lower expected returns if the short‐sellers have more wealth, lower risk aversion, or shorter horizon. The pricing of liquidity risk is different for derivatives than for positive‐net‐supply assets, and depends on investors' net nontraded risk exposure. We estimate this model for the credit default swap market. We find strong evidence for an expected liquidity premium earned by the credit protection seller. The effect of liquidity risk is significant but economically small.     

Utility valuation of multi-name credit derivatives and application to CDOs

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyse resulting yield spreads in multi-name credit derivatives, particularly CDOs. We study first the idealized problem with constant intensities where solutions are essentially explicit. We also give the large portfolio asymptotics for this problem. We then analyse the case where the firms have stochastic default intensities driven by a common factor, which can be viewed as another extreme from the independent case. This involves the numerical solution of a system of reaction-diffusion PDEs. We observe that the nonlinearity of the utility-indifference valuation mechanism enhances the effective correlation between the times of the credit events of the various firms leading to non-trivial senior tranche spreads, as often seen from market data.     

Moody's correlated binomial default distributions for inhomogeneous portfolios

This paper generalizes Moody's correlated binomial default distribution for homogeneous (exchangeable) credit portfolios, which was introduced by Witt, to the case of inhomogeneous portfolios. We consider two cases of inhomogeneous portfolios. In the first case, we treat a portfolio whose assets have uniform default correlation and non-uniform default probabilities. We obtain the default probability distribution and study the effect of inhomogeneity. The second case corresponds to a portfolio with inhomogeneous default correlation. Assets are categorized into several different sectors and the inter-sector and intra-sector correlations are not the same. We construct the joint default probabilities and obtain the default probability distribution. We show that as the number of assets in each sector decreases, inter-sector correlation becomes more important than intra-sector correlation. We study the maximum values of the inter-sector default correlation. Our generalization method can be applied to any correlated binomial default distribution model that has explicit relations to the conditional default probabilities or conditional default correlations, e.g. Credit Risk⁺, implied default distributions. We also compare some popular CDO pricing models from the viewpoint of the range of the implied tranche correlation.     

Pricing distressed CDOs with stochastic recovery

In this article, a framework for the joint modelling of default and recovery risk in a portfolio of credit risky assets is presented. The model especially accounts for the correlation of defaults on the one hand and correlation of default rates and recovery rates on the other hand. Nested Archimedean copulas are used to model different dependence structures. For the recovery rates a very flexible continuous distribution with bounded support is applied, which allows for an efficient sampling of the loss process. Due to the relaxation of the constant 40% recovery assumption and the negative correlation of default rates and recovery rates, the model is especially suited for distressed market situations and the pricing of super senior tranches. A calibration to CDO tranche spreads of the European iTraxx portfolio is performed to demonstrate the fitting capability of the model. Applications to delta hedging as well as base correlations are presented.     

Counterparty Risk for Credit Default Swaps: Markov Chain Interacting Intensities Model with Stochastic Intensity

We analyze the counterparty risk for credit default swaps using the Markov chain model of portfolio credit risk of multiple obligors with interacting default intensity processes. The default correlation between the protection seller and underlying entity is modeled by an increment in default intensity upon the occurrence of an external shock event. The arrival of the shock event is a Cox process whose stochastic intensity is assumed to follow an affine diffusion process with jumps. We examine how the correlated default risks between the protection seller and the underlying entity may affect the credit default premium in a credit default swap.     

From equity to default correlation with taxes

For fixed income investment, the preponderant risk is the clustering of defaults in the portfolio. Accurate prediction of such clustering depends on the knowledge of default correlation. We develop models with exogenous debt and endogenous debt to predict default correlations from equity correlations based on a self-consistent structural framework. We also examine how taxes affect the prediction of default correlations based on the two models. The empirical analysis shows that the corporate taxes tend to decrease default correlations, while personal taxes could increase or decrease default correlations. Our default correlation model with exogenous debt does a better job of predicting default correlations for high quality bonds, while the one with endogenous debt predicts more accurately for lower rated bonds. Our studies not only theoretically improve the modeling of default correlation in the structural setting but also shed new light on various aspects of default correlations and thereby help financial practitioners price credit derivatives more accurately and formulate more effective strategies to manage default risk of credit portfolios.     

Modelling Credit Risk for Innovative SMEs: the Role of Innovation Measures

Small-medium enterprises (SMEs) encounter financial constraints when they try to obtain credit from banks. These constraints are particularly severe for innovative SMEs. Thus, developing models for innovative SMEs that provide reliable estimates of their probabilities of default (PD) is important because the PDs can also serve as ratings. We examine the role of innovative assets such as patents in credit risk modelling due to their signaling value. Specifically, we add to a logit model two innovation-related variables in order to account for both the dimension and the value of the patent portfolio. Based on a unique data set of innovative SMEs with default years of 2005–2008, we show that, although the value of the patent portfolio always reduces the PD, its dimension reduces the firm’s riskiness only if coupled with an appropriate equity level.     

Empirical credit cycles and capital buffer formation

We model 1927–1997 US business failure rates using an unobserved components time series model. Clear evidence is found of cyclical behavior in default rates. We also detect significant longer term movements in default rates and default correlations. In a multi-year backtest experiment we show that accommodation of default rate dynamics has important consequences for credit risk capitalization requirements. Static or myopic variants of credit portfolio models miss significant periods of credit risk accumulation. Empirically congruent dynamic models by contrast provide more timely warning signals of credit risk build-up. In this way they may mitigate some of the pro-cyclicality concerns.     

Default risk premium and asset prices

We estimate a standard structural model of credit risk to draw insights about the premium demanded by investors for bearing default risk, using data on credit default swaps and market capitalization. We pin down the daily market value of assets for a set of non-financial firms and uncover cross-sectional heterogeneity in terms of the magnitude and time variation of the premium. By exploring the link between asset and default risk premia, we show that this heterogeneity closely depends on the relationship between the firm-specific market value of the assets and the business cycle.     

Modelling default contagion using multivariate phase-type distributions

We model dynamic credit portfolio dependence by using default contagion in an intensity-based framework. Two different portfolios (with ten obligors), one in the European auto sector, the other in the European financial sector, are calibrated against their market CDS spreads and the corresponding CDS-correlations. After the calibration, which are perfect for the banking portfolio, and good for the auto case, we study several quantities of importance in active credit portfolio management. For example, implied multivariate default and survival distributions, multivariate conditional survival distributions, implied default correlations, expected default times and expected ordered default times. The default contagion is modelled by letting individual intensities jump when other defaults occur, but be constant between defaults. This model is translated into a Markov jump process, a so called multivariate phase-type distribution, which represents the default status in the credit portfolio. Matrix-analytic methods are then used to derive expressions for the quantities studied in the calibrated portfolios.     

信用衍生产品隐含相关性结构研究

信用衍生产品在信用风险管理领域日益流行。其标的资产池的违约相关性结构在信用衍生产品定价、信用组合多元化和信用组合风险管理中具有重要作用。本文通过复制信用衍生产品中存在的隐含相关性微笑曲线现象,研究信用衍生产品标的信用的违约相关性结构。结果表明,本文使用的研究方法能够较好地给出标的信用合约间的违约相关性结构,同时,本文从风险管理角度研究了算法的有效性、计算效率和稳健性,并给出了相应的解释。     

Credit cycles and macro fundamentals

We use an intensity-based framework to study the relation between macroeconomic fundamentals and cycles in defaults and rating activity. Using Standard and Poor's U.S. corporate rating transition and default data over the period 1980–2005, we directly estimate the default and rating cycle from micro data. We relate this cycle to the business cycle, bank lending conditions, and financial market variables. In line with earlier studies, the macro variables appear to explain part of the default cycle. However, we strongly reject the correct dynamic specification of these models. The problem is solved by adding an unobserved dynamic component to the model, which can be interpreted as an omitted systematic credit risk factor. By accounting for this latent factor, many of the observed macro variables loose their significance. There are a few exceptions, but the economic impact of the observed macro variables for credit risk remains low. We also show that systematic credit risk factors differ over transition types, with risk factors for downgrades being noticeably different from those for upgrades. We conclude that portfolio credit risk models based only on observable systematic risk factors omit one of the strongest determinants of credit risk at the portfolio level. This has obvious consequences for current modeling and risk management practices.     

IRB法下公司敞口组合的风险驱动因子计量研究

巴塞尔银行监管委员会针对防范信贷组合信用风险所需要的资本制定的内部评级法,通过风险驱动因子的变化来反映组合回报的变化,并根据风险权重函数,通过风险加权资产转化为与每一项信用风险敞口更准确匹配的资本要求.本文对违约概率、违约损失率、违约敞口、期限因素以及违约相关性等信贷组合信用风险的风险驱动因子的度量进行了综合研究.     

Constant Proportion Debt Obligations (CPDOs): modeling and risk analysis

Constant Proportion Debt Obligations (CPDOs) are structured credit derivatives that generate high coupon payments by dynamically leveraging a position in an underlying portfolio of investment-grade index default swaps. CPDO coupons and principal notes received high initial credit ratings from the major rating agencies, based on complex models for the joint transition of ratings and spreads for all names in the underlying portfolio. We propose a parsimonious model for analysing the performance of CPDO strategies using a top-down approach that captures the essential risk factors of the CPDO. Our approach allows us to compute default probabilities, loss distributions and other tail risk measures for the CPDO strategy and analyse the dependence of these risk measures on various parameters describing the risk factors. We find that the probability of the CPDO defaulting on its coupon payments can be made arbitrarily small—and thus the credit rating arbitrarily high—by increasing leverage, but the ratings obtained strongly depend on assumptions on the credit environment (high spread or low spread). More importantly, CPDO loss distributions are found to exhibit a wide range of tail risk measures inside a given rating category, suggesting that credit ratings are insufficient performance indicators for such complex leveraged strategies. A worst-case scenario analysis indicates that CPDO strategies have a high exposure to persistent spread-widening scenarios and that CPDO ratings are shown to be quite unstable during the lifetime of the strategy.     

Credit derivatives with multiple debt issues

We evaluate the most actively traded types of credit derivatives within a unified pricing framework that allows for multiple debt issues. Since firms default on all of their obligations, total debt is instrumental in the likelihood of default and therefore in credit derivatives valuation. We use a single factor interest rate model where the exponential default frontier is based on total debt and is made coherent with observed bond prices. Analytical formulae are derived for credit default swaps, total return swaps (both fixed-for-fixed and fixed-for-floating), and credit risk options (CROs). Price behaviors and hedging properties of all these credit derivatives are investigated. Simulations document that credit derivatives prices may be significantly affected by terms of debt other than those of the reference obligation. The analysis of CROs indicates their superior ability to fine-tune the hedging of magnitude and arrival risks of default.     

Modelling the distribution of credit losses with observable and latent factors

This paper proposes a dynamic model to estimate the credit loss distribution of the aggregate portfolio of loans granted in a banking system. We consider a sectoral approach distinguishing between corporates and households. The evolution of their default frequencies and the size of the loans portfolio are expressed as functions of macroeconomic conditions as well as unobservable credit risk factors, which capture contagion effects between sectors. In addition, we model the distributions of the Exposures at Default and the Losses Given Default. We apply our framework to the Spanish banking system, where we find that sectoral default frequencies are not only affected by economic cycles but also by a persistent latent factor. Finally, we identify the riskier sectors, perform stress tests and compare the relative risk of small and large institutions.     

Bilateral credit valuation adjustment for large credit derivatives portfolios

We obtain an explicit formula for the bilateral counterparty valuation adjustment of a credit default swaps portfolio referencing an asymptotically large number of entities. We perform the analysis under a doubly stochastic intensity framework, allowing default correlation through a common jump process. The key insight behind our approach is an explicit characterization of the portfolio exposure as the weak limit of measure-valued processes associated with survival indicators of portfolio names. We validate our theoretical predictions by means of a numerical analysis, showing that counterparty adjustments are highly sensitive to portfolio credit risk volatility as well as to the intensity of the common jump process.