Credit risk modeling with affine processes

This article combines an orientation to credit risk modeling with an introduction to affine Markov processes, which are particularly useful for financial modeling. We emphasize corporate credit risk and the pricing of credit derivatives. Applications of affine processes that are mentioned include survival analysis, dynamic term-structure models, and option pricing with stochastic volatility and jumps. The default-risk applications include default correlation, particularly in first-to-default settings. The reader is assumed to have some background in financial modeling and stochastic calculus.     

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.     

Credit Derivatives in an Affine Framework

An efficient method for valuing credit derivatives based on three entities is developed in an affine framework. This includes interdependence of market and credit risk, joint credit migration and counterparty default risk of three firms. As an application we provide closed form expressions for the joint distribution of default times, default correlations, and default swap spreads in the presence of counterparty default risk. Vienna Institute of Finance is funded by WWTF (Vienna Science and Technology Fund).     

Capital Allocation for Portfolio Credit Risk

Capital allocation rules are derived that maximize leverage while maintaining a target solvency rate for credit portfolios where risk is driven by a single common factor and idiosyncratic risk is fully diversified. Equilibrium conditions ensure that capital allocations depend on interest earnings as well as credits’ probability of default, endogenous loss given default, and asset correlation. Capitalization rates exceed those estimated using Gaussian credit loss models. Results demonstrate that credit risk is undercapitalized by the Basel II AIRB approach in part because of ambiguities regarding the definition of loss given default. An alternative proposed capital rule removes this bias.     

The term structure of credit spreads with jump risk

Default risk analysis is important for valuing corporate bonds, swaps, and credit derivatives and plays a critical role in managing the credit risk of bank loan portfolios. This paper offers a theory to explain the observed empirical regularities on default probabilities, recovery rates, and credit spreads. It incorporates jump risk into the default process. With the jump risk, a firm can default instantaneously because of a sudden drop in its value. As a result, a credit model with the jump risk is able to match the size of credit spreads on corporate bonds and can generate various shapes of yield spread curves and marginal default rate curves, including upward-sloping, downward-sloping, flat, and hump-shaped, even if the firm is currently in a good financial standing. The model also links recovery rates to the firm value at default so that the variation in recovery rates is endogenously generated and the correlation between recovery rates and credit ratings before default reported in Altman [J. Finance 44 (1989) 909] can be justified.     

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.     

中小企业集合债券总体信用风险度量研究

中小企业集合债券总体信用风险既包括系统风险产生的周期性违约风险,又包括相互关联关系导致的传染性违约风险。首先通过对因素模型的改进构建模型Ⅰ,研究集合债券的周期性违约风险;在此基础上引入违约传染建立模型Ⅱ,分析违约传染对违约概率及违约相关性的影响,研究集合债券的总体信用风险。最后基于模型Ⅱ进行算例研究,得出结论：企业间的相互关联关系降低了其1次违约概率,增加了其多次违约概率即违约相关性。     

On cox processes and credit risky securities

A framework is presented for modeling defaultable securities and credit derivatives which allows for dependence between market risk factors and credit risk. The framework reduces the technical issues of modeling credit risk to the same issues faced when modeling the ordinary term structure of interest rates. It is shown how to generalize a model of Jarrow, Lando and Turnbull (1997) to allow for stochastic transition intensities between rating categories and into default. This generalization can handle contracts with payments explicitly linked to ratings. It is also shown how to obtain a term structure model for all different rating categories simultaneously and how to obtain an affine-like structure. An implementation is given in a simple one factor model in which the affine structure gives closed form solutions.     

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.     

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.     

Credit Portfolio Loss Forecasts for Economic Downturns

Recent studies find a positive correlation between default and loss given default rates of credit portfolios. In response, financial regulators require financial institutions to base their capital on 'Downturn' loss rates given default which are also known as Downturn LGDs. This article proposes a concept for the Downturn LGD which incorporates econometric properties of credit risk as well as the information content of default and loss given default models. The concept is compared to an alternative proposal by the Department of the Treasury, the Federal Reserve System and the Federal Insurance Corporation. An empirical analysis is provided for US American corporate bond portfolios of different credit quality, seniority and security.     

An Empirical Analysis of Personal Bankruptcy and Delinquency

This article uses a new dataset of credit card accounts to analyzecredit card delinquency, personal bankruptcy, and the stabilityof credit risk models. We estimate duration models for defaultand assess the relative importance of different variables inpredicting default. We investigate how the propensity to defaulthas changed over time, disentangling the two leading explanationsfor the recent increase in default rates—a deteriorationin the risk composition of borrowers versus an increase in borrowers'willingness to default due to declines in default costs. Evenafter controlling for risk composition and economic fundamentals,the propensity to default significantly increased between 1995and 1997. Standard default models missed an important time-varyingdefault factor, consistent with a decline in default costs.     

The analytic pricing of asymmetric defaultable swaps

Swaps where both parties are exposed to credit risk still lack convincing pricing mechanisms. This article presents a reduced-form model where the event of default is related to structural characteristics of each party. The cash flows submitted to credit risk are identified before the swap is priced. Analytical pricing formulas for interest rate and currency swaps are computed using a Gaussian model for risky bonds. Currency swaps exhibit additional correlation risk. The benefits from netting depend on the balance between exposures and market conditions in valuation. We show that sources of credit risk asymmetries are also likely to impact on credit spreads.     

Time‐Varying Asset Volatility and the Credit Spread Puzzle

Most extant structural credit risk models underestimate credit spreads—a shortcoming known as the credit spread puzzle. We consider a model with priced stochastic asset risk that is able to fit medium‐ to long‐term spreads. The model, augmented by jumps to help explain short‐term spreads, is estimated on firm‐level data and identifies significant asset variance risk premia. An important feature of the model is the significant time variation in risk premia induced by the uncertainty about asset risk. Various extensions are considered, among them optimal leverage and endogenous default.     

Can Listing Information Indicate Borrower Credit Risk in Online Peer-to-Peer Lending?

Effective assessment of borrower credit risk is the greatest challenge for peer-to-peer (P2P) lenders, especially in the Chinese market, where borrowers lack widely recognized credit scores. In this study, based on credit data from 2012 to 2015 from the website Renrendai.com, a logit model was used to assess borrower credit risk and predict the probability of default in every out-of-sample listing. The predicted probability of default was then compared with the actual default observation of default. The empirical results show that the logit model can evaluate the credit risk of P2P borrowers, and the model reduces the default rate to 9.5%, compared with the total sample default rate of 16.5%.     

The Cross‐Section of Credit Risk Premia and Equity Returns

We explore the link between a firm's stock returns and credit risk using a simple insight from structural models following Merton ( 1974 ): risk premia on equity and credit instruments are related because all claims on assets must earn the same compensation per unit of risk. Consistent with theory, we find that firms' stock returns increase with credit risk premia estimated from CDS spreads. Credit risk premia contain information not captured by physical or risk‐neutral default probabilities alone. This sheds new light on the “distress puzzle”—the lack of a positive relation between equity returns and default probabilities—reported in previous studies.     

Value investing in credit markets

We outline a parsimonious empirical model to assess the relative usefulness of accounting- and equity market-based information to explain corporate credit spreads. The primary determinant of corporate credit spreads is the physical default probability. We compare existing accounting-based and market-based models to forecast default. We then assess whether the credit market completely incorporates this default information into credit spreads. We find that credit spreads reflect information about forecasted default rates with a significant lag. This unique evidence suggests a role for value investing in credit markets.     

Modeling severity risk under PD–LGD correlation

In this article, a generic severity risk framework in which loss given default (LGD) is dependent upon probability of default (PD) in an intuitive manner is developed. By modeling the conditional mean of LGD as a function of PD, which also varies with systemic risk factors, this model allows an arbitrary functional relationship between PD and LGD. Based on this framework, several specifications of stochastic LGD are proposed with detailed calibration methods. By combining these models with an extension of CreditRisk+, a versatile mixed Poisson credit risk model that is capable of handling both risk factor correlation and PD–LGD dependency is developed. An efficient simulation algorithm based on importance sampling is also introduced for risk calculation. Empirical studies suggest that ignoring or incorrectly specifying severity risk can significantly underestimate credit risk and a properly defined severity risk model is critical for credit risk measurement as well as downturn LGD estimation.     

Credit Contagion from Counterparty Risk

Standard credit risk models cannot explain the observed clustering of default, sometimes described as "credit contagion." This paper provides the first empirical analysis of credit contagion via direct counterparty effects. We find that bankruptcy announcements cause negative abnormal equity returns and increases in CDS spreads for creditors. In addition, creditors with large exposures are more likely to suffer from financial distress later. This suggests that counterparty risk is a potential additional channel of credit contagion. Indeed, the fear of counterparty defaults among financial institutions explains the sudden worsening of the credit crisis after the Lehman bankruptcy in September 2008.     

A first-passage-time model under regime-switching market environment

In this paper, we suggest a first-passage-time model which can explain default probability and default correlation dynamics under stochastic market environment. We add a Markov regime-switching market condition to the first-passage-time model of Zhou [Zhou, C., 2001. An analysis of default correlations and multiple defaults. Review of Financial Studies 14, 555–576]. Using this model, we try to explain various relationship between default probability, default correlation, and market condition. We also suggest a valuation method for credit default swap (CDS) with (or without) counterparty default risk (CDR) and basket default swap under this model.Our numerical results provide us with several meaningful implications. First, default swap spread is higher in economic recession than in economic expansion across default swap maturity. Second, as the difference of asset return volatility between under bear market and under bull market increases, CDS spread increases regardless of maturity. Third, the bigger the intensity shifting from bull market to bear market, the higher the spread for both CDS without CDR and basket default swap.