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.     

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.     

Tranching and Rating

In this paper we analyse the source and magnitude of marketing gains from selling structured debt securities at yields that reflect only their credit ratings, or specifically at yields on equivalently rated corporate bonds. We distinguish between credit ratings that are based on probabilities of default and ratings that are based on expected default losses. We show that subdividing a bond issued against given collateral into subordinated tranches can yield significant profits under the hypothesised pricing system. Increasing the systematic risk or reducing the total risk of the bond collateral increases the profits further. The marketing gain is generally increasing in the number of tranches and decreasing in the rating of the lowest rated tranche.     

An analytical approach for systematic risk sensitivity of structured finance products

The global financial crisis has shown that many financial institutions dealing with credit derivatives were exposed to severe unexpected losses. This indicates that systematic influences are decisively underestimated particularly with regard to structured products like securitized tranches of collateralized debt obligations. Our analytical study addresses these systematic effects: We provide a simple model which allows a closed-form comparison of both bonds and tranches with respect to their systematic risk. We demonstrate that the exposure to systematic risk of tranches may be many times higher than the exposure of bonds, even if both products share the same rating grade, e.g., an ‘AAA’ rating, measured by either default probability or expected loss. Particularly in economic downturns, default rates of tranches may be multiples of those of bonds. Our results help understand high default rates of tranches during the financial crisis and show that classical ratings are insufficient metrics for measuring risks of structured products.     

Securitization bubbles: Structured finance with disagreement about default risk

An additional reason for the structured finance boom of the 2000s may have been disagreement about default risk of collateral assets. When risk-neutral investors disagree about average default probabilities, structuring collateral cash flow raises prices by concentrating optimists’ demand on risky tranches. With disagreement about default correlation, low-correlation investors believe in diversification and pay high prices for senior tranches they deem riskless. High-correlation investors value junior tranches they expect to pay whenever aggregate conditions are good. Risk aversion and short selling through credit default swaps reduce the prices of both pass-through and structured securitizations but may increase the return to tranching.     

中国债务抵押债券定价模型及实证分析

利用KMV模型估算出各债务人的违约概率,并用Copula函数分别估算出债务人之间的违约相关系数,模拟出各债务人的违约时点,在此基础上对债务抵押债券各系列进行定价;进一步应用Gaussian Copula和Student-t Copula对中国金融市场上发行的信贷抵押债券08招元一期产品各系列价差进行实证研究,结果发现两种Copula方法在估计优先A1级系列与B级系列时出现低估现象,而在估计优先A2级时出现高估;但一般都在5个基点的可接受误差范围之内。这表明该方法可以应用于信贷资产支持证券的定价,并可对高收益级的到期收益率进行估计。     

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.     

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.     

Competition in the credit rating Industry: Benefits for investors and issuers

We empirically investigate the benefits of multiple ratings not only at issuance of debt instruments but also during the subsequent monitoring phase. Using a record of monthly credit rating migration data on all U.S. residential mortgage-backed securities rated by Standard & Poor's, Moody's, and Fitch between 1985 and 2012 (154,600 tranches), our results provide empirical evidence that rating agencies put more effort in rating and outlook revisions when tranches have assigned multiple ratings. Furthermore, we see that in the case of multiple ratings, agencies do a better job in discriminating tranches with respect to default risk. On the downside, we observe a shift in collateral towards senior tranches and incentives for issuers to engage in rating shopping activities, but find no evidence that rating agencies exploit such behavior to attract more rating business. Our results contribute to the literature on information production of credit ratings and extend the perspective to the monitoring period after issuance.     

Pricing of credit default index swap tranches with one-factor heavy-tailed copula models

In this paper, we provide two one-factor heavy-tailed copula models for pricing a collateralized debt obligation and credit default index swap tranches: (1) a one-factor double t distribution with fractional degrees of freedom copula model and (2) a one-factor double mixture distribution of t and Gaussian distribution copula model. A time-varying tail-fatness parameter is introduced in each model, allowing one to change the tail-fatness of the copula function continuously. Fitting our model to comprehensive market data, we find that a model with fixed tail-fatness cannot fit market data well over time. The two models that we propose are capable of fitting market data well over time when using a proper time-varying tail-fatness parameter. Moreover, we find that the time-varying tail-fatness parameters change dramatically over a one-year period.     

CLO Performance

We study the performance of collateralized loan obligations (CLOs) to understand the market imperfections giving rise to these vehicles and their corresponding economic costs. CLO equity tranches earn positive abnormal returns from the risk-adjusted price differential between leveraged loans and CLO debt tranches. Debt tranches offer higher returns than similarly rated corporate bonds, making them attractive to banks and insurers that face risk-based capital requirements. Temporal variation in equity performance highlights the resilience of CLOs to market volatility due to their closed-end structure, long-term funding, and embedded options to reinvest principal proceeds.     

Pricing CDOs with state-dependent stochastic recovery rates

Up to the 2007 crisis, research within bottom-up CDO models mainly concentrated on the dependence between defaults. Since then, due to substantial increases in market prices of systemic credit risk protection, more attention has been paid to recovery rate assumptions. In this paper, we use stochastic orders theory to assess the impact of recovery on CDOs and show that, in a factor copula framework, a decrease of recovery rates leads to an increase of the expected loss on senior tranches, even though the expected loss on the portfolio is kept fixed. This result applies to a wide range of latent factor models and is not specific to the Gaussian copula model. We then suggest introducing stochastic recovery rates in such a way that the conditional on the factor expected loss (or, equivalently, the large portfolio approximation) is the same as in the recovery markdown case. However, granular portfolios behave differently. We show that a markdown is associated with riskier portfolios than when using the stochastic recovery rate framework. As a consequence, the expected loss on a senior tranche is larger in the former case, whatever the attachment point. We also deal with implementation and numerical issues related to the pricing of CDOs within the stochastic recovery rate framework. Due to differences across names regarding the conditional (on the factor) losses given default, the standard recursion approach becomes problematic. We suggest approximating the conditional on the factor loss distributions, through expansions around some base distribution. Finally, we show that the independence and comonotonic cases provide some easy to compute bounds on expected losses of senior or equity tranches.     

On the Relative Pricing of Long‐Maturity Index Options and Collateralized Debt Obligations

We investigate a structural model of market and firm‐level dynamics in order to jointly price long‐dated S&P 500 index options and CDO tranches of corporate debt. We identify market dynamics from index option prices and idiosyncratic dynamics from the term structure of credit spreads. We find that all tranches can be well priced out‐of‐sample before the crisis. During the crisis, however, our model can capture senior tranche prices only if we allow for the possibility of a catastrophic jump. Thus, senior tranches are nonredundant assets that provide a unique window into the pricing of catastrophic risk.     

Are All Ratings Created Equal? The Impact of Issuer Size on the Pricing of Mortgage‐Backed Securities

Initial yields on both AAA‐rated and non‐AAA rated mortgage‐backed security (MBS) tranches sold by large issuers are higher than yields on similar tranches sold by small issuers during the market boom years of 2004 to 2006. Moreover, the prices of MBS sold by large issuers drop more than those sold by small issuers, and the differences are concentrated among tranches issued during 2004 to 2006. These results suggest that investors price the risk that large issuers received more inflated ratings than small issuers, especially during boom periods.     

Omitted Mobility Characteristics and Property Market Dynamics: Application to Mortgage Termination

Property market dynamics depend on changes in long run equilibrium and on impediments to adjustment towards equilibrium. Mortgage termination due to mobility, default and refinancing provides a lens for evaluating property market adjustments. The borrower’s decision to move as an adjustment mechanism in the property market is associated with utility-maximizing decisions to either prepay or default on the mortgage. The optimal choice between these two termination events may depend on unobserved propensities related to change in income, job location, or family size, and substantial inertial forces including search costs, neighborhood change and attachment to an area. We propose a method for modeling variables determining the impact of mobility on mortgage terminations with imperfect household and loan level data. Since omitted variables contribute to moving decisions and therefore to mortgage prepayment and default decisions, utility functions for sale and default are correlated through these unobservable variables; thus, the IIA assumption of the widely used Multinomial Logit Model (MNL) is violated. Under such circumstances, econometric theory suggests that the Nested Logit Model (NMNL) is a better choice, which obviates the limitation of MNL by allowing correlation in unobserved factors across alternatives. Using loan level micro data, we find empirical evidence showing significant correlation between sale and default due to omitted borrower mobility characteristics. Our simulations find that NMNL out performs MNL in out-of-sample prediction. Improved predictions of moves and defaults are applicable to micro and macro analysis of the housing market system.     

Did investors outsource their risk analysis to rating agencies? Evidence from ABS-CDOs

Based on a sample of 3254 floating rate tranches from 617 ABS-CDOs (collateralized debt obligations backed by asset-backed securities), this paper tests the “rating overdependence” hypothesis – i.e., that ratings of structured products are a sufficient statistic (in terms of predicting future credit performance) for yield spreads at origination. The paper’s findings are fourfold. First, yield spreads at issuance predict future performance of ABS-CDO tranches even after controlling for the information contained in ratings. Second, the ability of yield spreads to predict future performance, however, is driven exclusively by ratings below AAA (and, to a lesser extent, also by the lowest priority AAA tranches), whereas spreads of super senior AAA tranches show no information content. Third, the predictive ability of yield spreads is lower for tranches from later vintages and for tranches from deals with more complex collateral pools. Fourth, the conditional correlation between ratings and spreads, in turn, is increasing in time and higher for tranches from complex deals. In sum, the evidence indicates that investors in (especially AAA) tranches from later and more complex deals have avoided performing costly due diligence on the securities they bought.     

A comparative study of the probability of default for global financial firms

This article presents a modification of Merton’s (1976) ruin option pricing model to estimate the implied probability of default from stock and option market prices. To test the model, we analyze all global financial firms with traded options in the US and focus on the subprime mortgage crisis period. We compare the performance of the implied probability of default from our model to the expected default frequencies based on the Moody’s KMV model and agency credit ratings by constructing cumulative accuracy profiles (CAP) and the receiver operating characteristic (ROC). We find that the probability of default estimates from our model are equal or superior to other credit risk measures studied based on CAP and ROC. In particular, during the subprime crisis our model surpassed credit ratings and matched or exceeded KMV in anticipating the magnitude of the crisis. We have also found some initial evidence that adding off-balance-sheet derivatives exposure improves the performance of the KMV model.     

A Top-Down Approach for Asset-Backed Securities: A Consistent Way of Managing Prepayment, Default and Interest Rate Risks

We define a new approach to manage prepayment, default and interest rate risks simultaneously in some standard asset-backed securities structures. We propose a parsimonious top-down approach, by modeling directly the portfolio loss process and the amortization process. Both are correlated to interest rates. The methodology is specified for sequential- and pro-rata pay bonds (ABS, CMO, CDO of ABS), cash or synthetic. We prove analytical formulas to price all tranches, under and without the simplifying assumption that amortization occurs in the most senior tranche only. The model behavior is illustrated through the empirical analysis of an actual synthetic ABS trade.     

Financial statement comparability and credit risk

Prior research shows that firms’ financial statement comparability improves the accuracy of market participants’ valuation judgments and thus may reduce firms’ costs of capital. Distinct from prior research focusing on the equity market, we develop measures of comparability relevant to debt market participants based on the within-industry variability of Moody’s adjustments to reported accounting numbers for the purposes of credit rating. We examine two sets of adjustments: (1) to the interest coverage ratio and (2) to non-recurring income items. We validate these comparability measures by providing evidence that greater comparability is associated with lower frequency and magnitude of split ratings by credit rating agencies. We predict and find that greater comparability is associated with (1) lower estimated bid-ask spreads for traded bonds, (2) lower credit spreads for both bonds and five-year credit default swaps, and (3) a steeper one- to five-year credit default swap term structure. Our results are consistent with financial statement comparability reducing debt market participants’ uncertainty about and pricing of firms’ credit risk.