Reduced Form Credit Risk Model Research Articles

Insider trading in an equilibrium model with default: a passage from reduced-form to structural modelling

We study, in the framework of Back [Rev. Financial Stud. 5(3), 387–409 (1992)], an equilibrium model for the pricing of a defaultable zero coupon bond issued by a firm. The market consists of a risk-neutral informed agent, noise traders, and a market maker who sets the price using the total order. When the insider does not trade, the default time possesses a default intensity in the market’s view as in reduced-form credit risk models. However, we show that, in equilibrium, the modelling becomes structural in the sense that the default time becomes the first time that some continuous observation process falls below a certain barrier. Interestingly, the firm value is still not observable. We also establish the no expected trade theorem that the insider’s trades are inconspicuous.

Recovery Rates in Credit Default Models Theory and Application to Corporate Bonds

In this paper I derive a reduced form credit risk model with stochastic recovery rates that is able to distinguish between default and bankruptcy and that is able to price defaulted debt consistently. First, I develop a general corporate bond pricing formula with the help of Characteristic Functions. Second, I specify the driving state variables as well as the functional form of intensities and recovery rates to derive a closed form valuation formula. In a numerical example I show how the resulting credit spreads depend on the default and bankruptcy parameters and the assumption of stochastic intensities and recovery rates. Credit spreads increase with higher default and bankruptcy intensities and decrease with higher recovery rates at default and at bankruptcy. If intensities and recovery rates are constant, the resulting term structure differs significantly. Additionally modeling the recovery rates stochastically has smaller effects.

Replication of Contingent Claims in a Reduced-Form Credit Risk Model with Discontinuous Asset Prices

The issue of replication of defaultable claims within the reduced-form approach to credit risk modeling and the related problem of completeness of intensity-based credit risk models are studied. Replication of survival claims under different assumptions on default intensities is examined in some detail. The considered model is fairly general with prices of traded assets governed by semimartingales.

The Delivery Option in Credit Default Swaps

Under standard assumptions the deterministic 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 the 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 trade at equal levels following default. Our extension models the implied expected recovery rate of the cheapest-to-deliver bond and, applied to the data, it largely eliminates the mispricing. The 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 the 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.

Default Correlation Dynamics with Business Cycle and Credit Quality Changes

The distinction between structural and reduced form credit risk models is now well known. Structural models arise from more compelling theoretical principles, but reduced form models are easier to implement in practice. As derivative instruments tied to portfolios of credit–sensitive securities, such as collateralized debt obligations (CDOs) and basket credit derivatives, become more common, the need to understand and model default correlations along with default probabilities also becomes important. But structural models have had a hard time producing the amount of correlation needed to match market prices for these products, partly because default intensities are typically modeled as being constant, or at most, functions of largely firm–specific variables. In this article, Kim and Kim introduce a tractable structural model in which default intensities also depend on broad macroeconomic variables that are assumed to follow correlated diffusions. This joint dependence on macro factors can induce strong default correlations, and can also help to explain certain results from the literature, like the fact that credit risk seems to be more highly correlated among high–grade firms than among low–grade firms, even though the latter have higher individual probabilities of default. The model can also produce a rich dynamic in credit risk exposure.

Modeling credit risk with partial information

This paper provides an alternative approach to Duffie and Lando [Econometrica 69 (2001) 633-664] for obtaining a reduced form credit risk model from a structural model. Duffie and Lando obtain a reduced form model by constructing an economy where the market sees the manager's information set plus noise. The noise makes default a surprise to the market. In contrast, we obtain a reduced form model by constructing an economy where the market sees a reduction of the manager's information set. The reduced information makes default a surprise to the market. We provide an explicit formula for the default intensity based on an Azema martingale, and we use excursion theory of Brownian motions to price risky debt.

An Empirical Study of Structural Credit Risk Models Using Stock and Bond Prices

Reduced-form credit risk models are often thought to be better suited than structural models for pricing corporate bonds. The authors challenge this view. Conditioned not only on equity but on bond and dividend information also, a structural model performs well compared to previously tested reduced-form models. In the pricing of bond portfolios, model errors are to a large extent diversifiable.

The effect of taxes on the pricing of defaultable debt

Empirical studies have documented the dependence of corporate credit spreads on default risk, equity premiums, and taxes. However, taxes have previously not been incorporated into reduced-form credit risk models. Therefore, we first extend the existing literature by considering a default intensity that depends on taxes as well as the default-free short rate and a market index. Consequently, we establish a theoretical basis to explain previous empirical findings regarding the significant impact of taxation on defaultable bond prices. Unlike previous models, tax implications for defaultable debt cannot be constructed from a sum of tax effects on zero coupon bonds. Our empirical tests then illustrate the importance of taxation. In particular, the impact of taxation increases as a function of the debt’s maturity and coupon rate.

Estimating expected losses and liquidity discounts implicit in debt prices

This paper provides an empirical implementation of a reduced form credit risk model that incorporates both liquidity risk and correlated defaults. Liquidity risk is modeled as a convenience yield and default correlation is modeled via an intensity process that depends on market factors. Various different liquidity risk and intensity process models are investigated. Firstly, the evidence supports a non-zero liquidity premium that is firm specific, reflecting idiosyncratic and not systematic risk. Secondly, the credit risk model with correlated defaults fits the data quite well with an average R of .87 and a pricing error of only 1.1 percent.

Bankruptcy Prediction With Industry Effects, Market Versus
 Accounting Variables, And Reduced Form Credit Risk Models

This paper investigates the forecasting accuracy of bankruptcy hazard rate models for U.S. companies over the time period 1962 - 1999 using both yearly and monthly observation intervals. The contribution of this paper is multiple-fold. One, using an expanded bankruptcy database we validate the superior forecasting performance of Shumway's (2001) model as opposed to Altman (1968) and Zmijewski (1984). Two, we demonstrate the importance of including industry effects in hazard rate estimation. Industry groupings are shown to significantly affect both the intercept and slope coefficients in the forecasting equations. Three, we extend the hazard rate model to apply to financial firms and monthly observation intervals. Due to data limitations, most of the existing literature employs only yearly observations. We show that bankruptcy prediction is markedly improved using monthly observation intervals. Fourth, consistent with the notion of market efficiency with respect to publicly available information, we demonstrate that accounting variables add little predictive power when market variables are already included in the bankruptcy model.