Copula dynamics in CDOs

Values of tranche spreads of collateralized debt obligations (CDOs) are driven by the joint default performance of the assets in the collateral pool. The dependence between the entities in the portfolio mainly depends on current economic conditions. Therefore, a correlation implied from tranches can be seen as a measure of the general situation of the credit market. We analyse the European market of standardized CDOs using tranches of the iTraxx index in the periods before and during the global financial crisis. We investigate the evolution of the correlations using different copula models: the standard Gaussian, the NIG, the double-t, and the Gumbel copula model. After calibration of these models, one obtains a time varying vector of parameters. We analyse the dynamic pattern of these coefficients. That enables us to forecast future parameters and consequently calculate Value-at-Risk measures for iTraxx Europe tranches.     

CDOs and the financial crisis: Credit ratings and fair premia

We study risk and return characteristics of CDOs using the market standard models. We find that fair spreads on CDO tranches are much higher than fair spreads on similarly-rated corporate bonds. Our results imply that credit ratings are not sufficient for pricing, which is surprising given their central role in structured finance markets. This illustrates limitations of the rating methodologies that are solely based on real-world default probabilities or expected losses and do not capture risk premia. We also demonstrate that CDO tranches have large exposure to systematic risk and thus their ratings and prices are likely to decline substantially when credit conditions deteriorate.     

Fast simulations in credit risk

We consider the problem of simulating tail loss probabilities and expected losses conditioned on exceeding a large threshold (expected shortfall) for credit portfolios. Our new idea, called the geometric shortcut, allows an efficient simulation for the case of independent obligors. It is even possible to show that, when the average default probability tends to zero, its asymptotic efficiency is higher than that of the naive algorithm. The geometric shortcut is also useful for models with dependent obligors and can be used for dependence structures modeled with arbitrary copulae. The paper contains the details for simulating the risk of the normal copula credit risk model by combining outer importance sampling with the geometric shortcut. Numerical results show that the new method is efficient in assessing tail loss probabilities and expected shortfall for credit risk portfolios. The new method outperforms all known methods, especially for credit portfolios consisting of weakly correlated obligors and for evaluating the tail loss probabilities at many thresholds in a single simulation run.     

Correlation smile matching for collateralized debt obligation tranches with α-stable distributions and fitted Archimedean copula models

As an extension of the standard Gaussian copula model to price collateralized debt obligation (CDO) tranche swaps we present a generalization of a one-factor copula model based on stable distributions. For special parameter values these distributions coincide with Gaussian or Cauchy distributions, but changing the parameters allows a continuous deformation away from the Gaussian copula. All these factor copulas are embedded in a framework of stochastic correlations. We furthermore generalize the linear dependence in the usual factor approach to a more general Archimedean copula dependence between the individual trigger variable and the common latent factor. Our analysis is carried out on a non-homogeneous correlation structure of the underlying portfolio. CDO tranche market premia, even throughout the correlation crisis in May 2005, can be reproduced by certain models. From a numerical perspective, all these models are simple, since calculations can be reduced to one-dimensional numerical integrals.     

Portfolio credit-risk optimization

This paper evaluates several alternative formulations for minimizing the credit risk of a portfolio of financial contracts with different counterparties. Credit risk optimization is challenging because the portfolio loss distribution is typically unavailable in closed form. This makes it difficult to accurately compute Value-at-Risk (VaR) and expected shortfall (ES) at the extreme quantiles that are of practical interest to financial institutions. Our formulations all exploit the conditional independence of counterparties under a structural credit risk model. We consider various approximations to the conditional portfolio loss distribution and formulate VaR and ES minimization problems for each case. We use two realistic credit portfolios to assess the in- and out-of-sample performance for the resulting VaR- and ES-optimized portfolios, as well as for those which we obtain by minimizing the variance or the second moment of the portfolio losses. We find that a Normal approximation to the conditional loss distribution performs best from a practical standpoint.     

Contagion models a la carte: which one to choose?

We propose a copula contagion mixture model for correlated default times. The model includes the well-known factor, copula, and contagion models as its special cases. The key advantage of such a model is that we can study the interaction of different models and their pricing impact. Specifically, we model the default times of the underlying names in a reference portfolio to follow contagion intensity processes with exponential decay coupled with a copula dependence structure. We also model the default time of the counterparty and its dependence structure with the reference portfolio. Numerical tests show that correlation and contagion have an enormous joint impact on the rates of CDO tranches and the corresponding credit value adjustments are extremely high to compensate for the wrong-way risk.     

Diversification in financial and crypto markets

This article investigates the conditional value at risk (CVaR) of two portfolio optimiza- tion approaches containing assets from the financial and crypto markets. We first catch the conditional interdependence structure among each variable through the vine-copula-GARCH model before merging it with the Mean-CVaR model. We then optimize each portfolio and find out the optimal allocation while evaluating the precise risk. The results indicate that the D-Vine copula is more suitable for both portfolios and that, when different conditional stock indices information are being taken into consideration, the crypto-market components can act as a weak hedge/safe haven against financial market indices. Furthermore, as CVaR is found to outperform the mean-variance of Markowitz in both portfolios, both risk measures similarly show that when including cryptocurrencies in a portfolio, the S&P 500 shall not be included. Additionally, the inclusion of Ethereum in a portfolio already containing Bitcoin does not boost the return.     

Portfolio selection with commodities under conditional copulas and skew preferences

This article investigates the portfolio selection problem of an investor with three-moment preferences taking positions in commodity futures. To model the asset returns, we propose a conditional asymmetric t copula with skewed and fat-tailed marginal distributions, such that we can capture the impact on optimal portfolios of time-varying moments, state-dependent correlations, and tail and asymmetric dependence. In the empirical application with oil, gold and equity data from 1990 to 2010, the conditional t copulas portfolios achieve better performance than those based on more conventional strategies. The specification of higher moments in the marginal distributions and the type of tail dependence in the copula has significant implications for the out-of-sample portfolio performance.     

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.     

基于藤结构Copula的多元信用风险相关性度量模型及其比较

在信贷组合管理的框架下,以行业信用风险为基础,构建了基于藤结构Copula的多元信用风险相关性度量模型,并以2006年6月～2010年12月中国上市公司数据对模型进行了参数估计,发现Canoni-cal藤结构更适合度量行业信用风险相关性,以电力煤气及水的生产为代表的强周期性行业对信用风险起着引导作用。     

Tests of conditional asset pricing models in the Brazilian stock market

In this paper, we test a version of the conditional CAPM with respect to a local market portfolio, proxied by the Brazilian stock index during the period 1976–1992. We also test a conditional APT model by using the difference between the 30-day rate (Cdb) and the overnight rate as a second factor in addition to the market portfolio in order to capture the large inflation risk present during this period. The conditional CAPM and APT models are estimated by the Generalized Method of Moments (GMM) and tested on a set of size portfolios created from individual securities exchanged on the Brazilian markets. The inclusion of this second factor proves to be important for the appropriate pricing of the portfolios.     

Cyclical correlations, credit contagion, and portfolio losses

We model aggregate credit losses on large portfolios of financial positions contracted with firms subject to both cyclical default correlation and direct default contagion processes. Cyclical correlation is due to the dependence of firms on common economic factors. Contagion is associated with the local interaction of firms with their business partners. We provide an explicit normal approximation of the distribution of portfolio losses. We quantify the relation between the variability of global economic fundamentals, strength of local firm interaction, and the fluctuation of losses. We find that cyclical oscillations in fundamentals dominate average losses, while local interaction causes additional fluctuations of losses around their average. The strength of the contagion-induced loss variability depends on the complexity of the business partner network.     

Haar wavelets-based approach for quantifying credit portfolio losses

This paper proposes a new methodology to compute Value at Risk (VaR) for quantifying losses in credit portfolios. We approximate the cumulative distribution of the loss function by a finite combination of Haar wavelet basis functions and calculate the coefficients of the approximation by inverting its Laplace transform. The Wavelet Approximation (WA) method is particularly suitable for non-smooth distributions, often arising in small or concentrated portfolios, when the hypothesis of the Basel II formulas are violated. To test the methodology we consider the Vasicek one-factor portfolio credit loss model as our model framework. WA is an accurate, robust and fast method, allowing the estimation of the VaR much more quickly than with a Monte Carlo (MC) method at the same level of accuracy and reliability.     

Hedging structured credit products during the credit crisis: A horse race of 10 models

Pricing and hedging structured credit products poses major challenges to financial institutions. This paper puts several valuation approaches through a crucial test: How did these models perform in one of the worst periods of economic history, September 2008, when Lehman Brothers went under? Did they produce reasonable hedging strategies? We study several bottom-up and top-down credit portfolio models and compute the resulting delta hedging strategies using either index contracts or a portfolio of single-name CDS contracts as hedging instruments. We compute the profit-and-loss profiles and assess the performances of these hedging strategies. Among all 10 pricing models that we consider the Student-t copula model performs best. The dynamical generalized-Poisson loss model is the best top-down model, but this model class has in general problems to hedge equity tranches. Our major finding is however that single-name and index CDS contracts are not appropriate instruments to hedge CDO tranches.     

Valuation of synthetic CDOs

In this paper, we consider the valuation of a synthetic collateralized debt obligation (CDO), a pool of underlying credit risky securities, “partitioned” into several tranches, each of which absorbs losses in accordance with its size and seniority. We derive a closed-form solution for credit spreads of the tranches of homogeneous pools and find an approximation for the credit spreads of inhomogeneous pools. The method leads to an accurate estimation of the credit spreads of synthetic CDOs and can be used in risk management applications.     

Determinants of asymmetric return comovements of gold and other financial assets

Using conditional time-varying copula models, we characterize the dependence structure of return comovements of gold and other financial assets (stocks, bonds, real estate and oil) during economic expansion and contraction regimes. We also investigate which key macroeconomic and non-macroeconomic variables significantly impact the asset return comovements using a two stage Markov Switching Stochastic Volatility (MSSV) framework. Our results show that the non-macro variables have significant influence on the return comovements. We find that gold is an inappropriate hedge against interest rate changes for real-estate and oil-based portfolios, while for bond portfolios, gold offers a good hedge against inflation uncertainty. We also provide evidence that the “flight to safety” phenomenon is due to the implied volatility of the stock market, rather than the observed stock market uncertainty. Finally, we forecast the asset return comovements and examine their economic significance. We show that a dynamic MSSV model which includes the macroeconomic and non-macroeconomic variables yields superior forecast of future asset return comovements when compared with a multivariate conditional covariance model.     

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.     

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.     

Estimating Portfolio Credit Losses in Downturns

This paper suggests formulas able to capture potential strong connection among credit losses in downturns without assuming any specific distribution for the variables involved. We first show that the current model adopted by regulators (Basel) is equivalent to a conditional distribution derived from the Gaussian Copula (which does not identify tail dependence). We then use conditional distributions derived from copulas that express tail dependence (stronger dependence across higher losses) to estimate the probability of credit losses in extreme scenarios (crises). Next, we use data on historical credit losses incurred in American banks to compare the suggested approach to the Basel formula with respect to their performance when predicting the extreme losses observed in 2009 and 2010. Our results indicate that, in general, the copula approach outperforms the Basel method in two of the three credit segments investigated. The proposed method is extendable to other differentiable copula families and this gives flexibility to future practical applications of the model.