A set-valued Markov chain approach to credit default

We propose a novel credit default model that takes into account the impact of macroeconomic factors and intergroup contagion on the defaults of obligors. We use a set-valued Markov chain to model the default process, which includes all defaulted obligors in the group. We obtain analytic characterizations for the default process and derive pricing formulas in explicit forms for synthetic collateralized debt obligations (CDOs). Furthermore, we use market data to calibrate the model and conduct numerical studies on the tranche spreads of CDOs. We find evidence to support that systematic default risk coupled with default contagion could have the leading component of the total default risk.     

Asymptotic Analysis of the Loss Given Default in the Presence of Multivariate Regular Variation

Consider a portfolio of n obligors subject to possible default. We propose a new structural model for the loss given default, which takes into account the severity of default. Then we study the tail behavior of the loss given default under the assumption that the losses of the n obligors jointly follow a multivariate regular variation structure. This structure provides an ideal framework for modeling both heavy tails and asymptotic dependence. Multivariate models involving Archimedean copulas and mixtures are revisited. As applications, we derive asymptotic estimates for the value at risk and conditional tail expectation of the loss given default and compare them with the traditional empirical estimates.     

Effects of the New Basel Capital Accord on Bank Capital Requirements for SMEs

Using data from three countries (US, Italy and Australia) and surveying related studies from several other countries in Europe, we investigate the effects of the New Basel Capital Accord on bank capital requirements for small and medium sized enterprises (SMEs). We find that, for all the countries, banks will have significant benefits, in terms of lower capital requirements, when considering small and medium sized firms as retail customers. But they will be obliged to use the Advanced IRB approach and to manage them on a pooled basis. For SMEs as corporate, however, capital requirements will be slightly greater than under the existing Basel I Capital Accord. We believe that most eligible banks will use a blended approach (considering some SMEs as retail and some as corporate). Through a breakeven analysis, we find that for all of our countries, banking organizations will be obliged to classify as retail at least 20% of their SME portfolio in order to maintain the current capital requirement (8%). JEL classification: G21, G28     

Cyclical default and recovery in stress testing loan losses

We present a macro variable-based empirical model for corporate bank loans’ credit risk. The model captures the well-known positive relationship between probability of default (PD) and loss given default (LGD; i.e., the inverse of recovery) and their counter-cyclical movement with the business cycle. In the absence of proper micro data on LGD, we use a random-sampling method to estimate the annual average LGD. We specify a two equation model for PD and LGD which is estimated with Finnish time-series data from 1989 to 2008. We also use a system of time-series models for the exogenous macro variables to derive the main macroeconomic shocks which are then used in stress testing aggregate loan losses. We show that the endogenous LGD makes a considerable difference in stress tests compared to a constant LGD assumption.     

Endogenous liquidity in credit derivatives

We study the determination of liquidity provision in the single-name credit default swap (CDS) market as measured by the number of distinct dealers providing quotes. We find that liquidity is concentrated among large obligors and those near the investment-grade/speculative-grade cutoff. Consistent with endogenous liquidity provision by informed financial institutions, more liquidity is associated with obligors for which there is a greater information flow from the CDS market to the stock market ahead of major credit events. Furthermore, the level of information heterogeneity plays an important role in how liquidity provision responds to transaction demand and how liquidity is priced into the CDS premium.     

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.     

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.     

The delivery option in credit default swaps

Under standard assumptions the reduced-form credit risk model is not capable of accurately pricing the two fundamental credit risk instruments – bonds and credit default swaps (CDS) – simultaneously. Using a data set of euro-denominated corporate bonds and CDS our paper quantifies this mispricing by calibrating such a model to bond data, and subsequently using it to price CDS, resulting in model CDS spreads up to 50% lower on average than observed in the market. An extended model is presented which includes the delivery option implicit in CDS contracts emerging since a basket of bonds is deliverable in default. By using a constant recovery rate standard models assume equal recoveries for all bonds and hence zero value for the delivery option. Contradicting this common assumption, case studies of Chapter 11 filings presented in the paper show that corporate bonds do not necessarily trade at equal levels following default. Our extension models the implied expected recovery rate of the cheapest-to-deliver bond and, applied to data, largely eliminates the mispricing. Calibrated recovery values lie between 8% and 47% for different obligors, exhibiting strong variation among rating classes and industries. A cross-sectional analysis reveals that the implied recovery parameter depends on proxies for the delivery option, primarily the number of available bonds and bond pricing errors. No evidence is found for a direct influence of the bid-ask spread, notional amount, coupon, or rating used as proxies for bond market liquidity.     

Features selection,data mining and finacial risk classification: a comparative study

The aim of this paper is to compare several predictive models that combine features selection techniques with data mining classifiers in the context of credit risk assessment in terms of accuracy, sensitivity and specificity statistics. The t‐statistic, Battacharrayia statistic, the area between the receiver operating characteristic, Wilcoxon statistic, relative entropy, and genetic algorithms were used for the features selection task. The selected features are used to train the support vector machine (SVM) classifier, backpropagation neural network, radial basis function neural network, linear discriminant analysis and naive Bayes classifier. Results from three datasets using a 10‐fold cross‐validation technique showed that the SVM provides the best accuracy under all features selections techniques adopted in the study for all three datasets. Therefore, the SVM is an attractive classifier to be used in real applications for bankruptcy prediction in corporate finance and financial risk management in financial institutions. In addition, we found that our best results are superior to earlier studies on the same datasets.     

Do measures of systemic risk predict U.S. corporate bond default rates?

Using univariate and multivariate Mixed Data Sampling (MIDAS) and LASSO estimation methodologies, we explore whether the U.S. annual average corporate bond default rate can be predicted by 12 monthly systemic risk measures proposed in the literature. We find that nearly all of the systemic risk indicators have predictive power for the default rate. Granger causality tests based on multivariate mixed frequency VAR models further support this conclusion. On the basis of MIDAS models, we illustrate that five of these indicators are able to forecast out-of-sample the 2009 corporate default crisis. Using a LASSO multivariate model, it is further shown that the systemic risk indicators can forecast out-of-sample both the 2009 default rate and the default rates during the buildup before the crisis and in the aftermath of the crisis. Institution-specific and volatility systemic risk measures are the most relevant for modeling U.S. corporate bond default rates, with the Conditional VaR measure of Adrian and Brunnermeier (2016) exhibiting the best performance.     

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.     

A simulation study of Basel II expected loss distributions for a portfolio of credit cards

Credit scoring models have been used traditionally as the basis of decisions to reject or accept credit applications. They are also used to categorize applicants or existing accounts into risk groups. Based on estimates of probability of default (PD), the risk groups may seem well separated. However, by considering distributions on risk elements such as model estimation uncertainty, exposure at default and loss given default, a simulation approach is used to compute Basel II expected loss distributions for a portfolio of credit cards. These show that discrimination between risk groups is not as clear as is immediately suggested simply by PD estimates. Based on these distributions, we also show that measuring extreme credit risk with Value at Risk can lead to considerable underestimation if distributions on these risk elements are not entered into the computation.     

Incorporating the dynamics of leverage into default prediction

A firm’s current leverage ratio is one of the core characteristics of credit quality used in statistical default prediction models. Based on the capital structure literature, which shows that leverage is mean-reverting to a target leverage, we forecast future leverage ratios and include them in the set of default risk drivers. An out-of-sample analysis of default predictions from a hazard model reveals that the discriminative power increases substantially when leverage forecasts are included. We further document that credit ratings contain information beyond the one contained in standard variables but that this information is unrelated to forecasts of leverage ratios.     

A Multifactor Model of Credit Spreads

We represent credit spreads across ratings as a function of common unobservable factors of the Vasicek form. Using a state-space approach we estimate the factors, their process parameters, and the exposure of each observed credit spread series to each factor. We find that most of the systematic variation across credit spreads is captured by three factors. The factors are closely related to the implied volatility index (VIX), the long bond rate, and S&P500 returns, supporting the predictions of structural models of default at an aggregate level. By making no prior assumption about the determinants of yield spread dynamics, our study provides an original and independent test of theory. The results also contribute to the current debate about the role of liquidity in corporate yield spreads. While recent empirical literature shows that the level and time-variation in corporate yield spreads is driven primarily by a systematic liquidity risk factor, we find that the three most important drivers of yield spread levels relate to macroeconomic variables. This suggests that if credit spread levels do contain a large liquidity premium, the time variation of this premium is likely driven by the same factors as default risk.     

Estimating default barriers from market information

Brockman and Turtle [J. Finan. Econ., 2003, 67, 511–529] develop a barrier option framework to show that default barriers are significantly positive. Most implied barriers are typically larger than the book value of corporate liabilities. We show theoretically and empirically that this result is biased due to the approximation of the market value of corporate assets by the sum of the market value of equity and the book value of liabilities. This approximation leads to a significant overestimation of the default barrier. To eliminate this bias, we propose a maximum likelihood (ML) estimation approach to estimate the asset values, asset volatilities, and default barriers. The proposed framework is applied to empirically examine the default barriers of a large sample of industrial firms. This paper documents that default barriers are positive, but not very significant. In our sample, most of the estimated barriers are lower than the book values of corporate liabilities. In addition to the problem with the default barriers, we find significant biases on the estimation of the asset value and the asset volatility of Brockman and Turtle.     

Modeling default risk with support vector machines

Predicting default risk is important for firms and banks to operate successfully. There are many reasons to use nonlinear techniques for predicting bankruptcy from financial ratios. Here we propose the so-called Support Vector Machine (SVM) to predict the default risk of German firms. Our analysis is based on the Creditreform database. In all tests performed in this paper the nonlinear model classified by SVM exceeds the benchmark logit model, based on the same predictors, in terms of the performance metric, AR. The empirical evidence is in favor of the SVM for classification, especially in the linear non-separable case. The sensitivity investigation and a corresponding visualization tool reveal that the classifying ability of SVM appears to be superior over a wide range of SVM parameters. In terms of the empirical results obtained by SVM, the eight most important predictors related to bankruptcy for these German firms belong to the ratios of activity, profitability, liquidity, leverage and the percentage of incremental inventories. Some of the financial ratios selected by the SVM model are new because they have a strong nonlinear dependence on the default risk but a weak linear dependence that therefore cannot be captured by the usual linear models such as the DA and logit models.     

Rapid and accurate development of prices and Greeks for nth to default credit swaps in the Li model

New techniques are introduced for pricing nth to default credit swaps in the Li model. We demonstrate the use of importance sampling to greatly increase the rate of convergence of Monte Carlo simulations for pricing. This technique is combined with the likelihood ratio and pathwise methods for computing the sensitivities of these products to changes in the hazard rates of the underlying obligors. In particular the extension of the pathwise method has wider significance in that it is shown that the method can be used even when the pay-off is discontinuous.     

Using a hybrid evolution approach to forecast financial failures for Taiwan-listed companies

Bankruptcy has been an important topic in finance and accounting research for a long time. Recent major bankruptcies have included seemingly robust companies such as Enron, Kmart, Global Crossing, WorldCom, and Lehman Brothers. These cases have become of serious public concern due to the huge influence these companies have on the real economy. This research proposes a hybrid evolution approach to integrate particle swarm optimization (PSO) with the support vector machine (SVM) technique for the purpose of predicting financial failures. The preparation phase collected an initial sample of 68 companies listed by the Taiwan Stock Exchange Corporation (TSEC). The financial datasets were constructed based on 33 financial ratios, four non-financial ratios and one combined macroeconomic index. To select suitable indicators for the input vector, the principle component analysis (PCA) technique was applied to reduce the data and determine how groupings of indicators measure the same concept. In the swarming phase, PSO was applied to obtain suitable parameters for SVM modeling without reducing the classification accuracy rate. In the modeling phase, the SVM model was used to build a training set that was used to calculate the model's accuracy and fitness value. Finally, these optimized parameters were used in the hybrid PSO–SVM model to evaluate the model's predictive accuracy. This paper provides four critical contributions. (1) Using the PCA technique, the statistical results indicate that the financial prediction performance is mainly affected by financial ratios rather than non-financial and macroeconomic ratios. (2) Even with the input of nearly 70% fewer indicators, our approach is still able to provide highly accurate forecasts of financial bankruptcy. (3) The empirical results show that the PSO–SVM model provides better classification accuracy (i.e. normal vs. bankrupt) than the grid search (Grid–SVM) approach. (4) For six well-known UCI datasets, the PSO–SVM model also provides better prediction accuracy than the Grid–SVM, GA–SVM, SVM, SOM, and SVR–SOM approaches. Therefore, this paper proposes that the PSO–SVM approach is better suited for predicting potential financial distress.     

The Mills Ratio and the behavior of redeemable bond prices in the Gaussian structural model of corporate default

This paper shows that forward default intensities in the Black and Cox (1976) model of corporate default can be expressed in terms of the Mills Ratio (Mills, 1926). The behaviour of the forward default intensity and hence the survivorship functions then follows from inequalities that are satisfied by this ratio. This allows me to analyze the effect of the firm’s distance to default, growth rate and volatility upon the value of its debt. These results can be used to analyze the comparative static properties of other models of corporate default and perhaps other first passage time models.     

Pricing Sovereign Credit Risk of Poland: Evidence from the CDS Market

We analyze the market assessment of sovereign credit risk using a reduced-form model to price the credit default swap (CDS) spreads, thus enabling us to derive values for the probability of default (PD) and loss given default (LGD) from the quotes of sovereign CDS contracts. We compare different specifications of the models allowing for both fixed and time-varying LGD, and we use these values to analyze the sovereign credit risk of Polish debt throughout the period of a global financial crisis. Our results suggest the presence of a low LGD and a relatively high PD during a recent financial crisis.