Term Structure Of Default Probabilities Research Articles

The Term Structure of Default Probabilities

Accounting standards require that financial institutions must measure default risk with respect to the full maturity of a financial instrument. This requires forecasting of future default probabilities. The forecast of future default probabilities concerns two aspects: forecasting macroeconomic scenarios and future average (with respect to the macro-economy) default probabilities. The present paper addresses the modelling of future average default probabilities. Due to the small number of defaults and observations this poses a difficult problem in the corporate area and requires a parsimonious model to deal with the sparse data situation. The degree of difficulty is made greater by the fact that default probabilities change for corporations via different patterns over time, with the pattern being depend on the initial rating grade. For initial investment-grade ratings the risk of a default increases, while for the bottom of the rating scale default probabilities decrease. To model these different patterns the paper proposes to extend the existing discrete-time survival model and to incorporate an additional time- and co-variate-dependent shape parameter into the hazard function. Using data from Standard & Poor's the paper shows that the shape parameter is able to reproduce the different patterns. The proposed model is bench-marked against a model which does not employ a shape parameter and the results show that the shape parameter improves in-sample and out-of-sample prediction.

Decomposing Rating Migration Matrices from Market Prices Revisited

The study revisits the method and premises of decomposing rating migration matrices from price data. Significantly easier, and more accurate method is presented. As an intermediary step, default and non-default probability term structures are first decomposed from market prices. This represents the composite default probability structure or process of a rating category, when considering rating migration and future ratings probabilities. A rating migration matrix is subsequently decomposed from the decomposed default probability term structures. Output residuals are low, yet the optimization problem finds many local solutions, rather than a solid global solution: different rating migration matrices can result in and produce the same decomposed default probability term structures. Furthermore, it is shown that this is not a matter of insufficient price data. Additional price data cannot solve this, and push the outcome to a global solution. The main implications are twofold: First, the underlying issue is that rating migration matrices state and reveal 2 components: a) default and non-default probability term structures, and b) non-default probability term sub-structures. The latter refers to a break-down of future non-default probabilities, per rating category – future rating probabilities in essence. Price data contain sufficient information to decompose the former, but not the latter. It is generally not possible to make non-default probability sub-structures (future rating probabilities) relevant in some way – to relate it to certain data – to allow decomposition, because there is little (market) data pertaining to this. The implication is that bond prices are not dependent on future ratings, over and above non-default probability term structures (“survival rates”), as a mirror of default probability term structures, as the latter already contains all price-relevant information. Secondly, rating migration matrices can be decomposed accurately enough, to provide some insight. At the same time, it may be unnecessary to decompose rating migration matrices, post default probability term structures. The latter can be done accurately and easily, and contain all the relevant price information.

Empirical Credit Risk Ratings of Individual Corporate Bonds and Derivation of Term Structures of Default Probabilities

Undoubtedly, it is important to have an empirically effective credit risk rating method for decision-making in the financial industry, business, and even government. In our approach, for each corporate bond (CB) and its issuer, we first propose a credit risk rating (Crisk-rating) system with rating intervals for the standardized credit risk price spread (S-CRiPS) measure presented by Kariya et al. (2015), where credit information is based on the CRiPS measure, which is the difference between the CB price and its government bond (GB)-equivalent CB price. Second, for each Crisk-homogeneous class obtained through the Crisk-rating system, a term structure of default probability (TSDP) is derived via the CB-pricing model proposed in Kariya (2013), which transforms the Crisk level of each class into a default probability, showing the default likelihood over a future time horizon, in which 1545 Japanese CB prices, as of August 2010, are analyzed. To carry it out, the cross-sectional model of pricing government bonds with high empirical performance is required to get high-precision CRiPS and S-CRiPS measures. The effectiveness of our GB model and the S-CRiPS measure have been demonstrated with Japanese and United States GB prices in our papers and with an evaluation of the credit risk of the GBs of five countries in the EU and CBs issued by US energy firms in Kariya et al. (2016a, b). Our Crisk-rating system with rating intervals is tested with the distribution of the ratings of the 1545 CBs, a specific agency’s credit rating, and the ratings of groups obtained via a three-stage cluster analysis.

Rating Migration and Bond Valuation: Towards Ahistorical Rating Migration Matrices and Default Probability Term Structures

The study examines rating migration, and default probability term structures obtained from rating migration matrices. It expands on the use of rating migration matrices with reduced form bond valuation models, by formally delineating the probability of default according to the likely rating paths of a bond, as implied by the rating migration matrix. Further, two alternatives are also considered. First, the cost of default is stipulated as the recovery of par according to the exit rating upon default. Also, in addition to stating the value of a bond in terms of expected cash flows, when considering the probability of default, the value of a bond is alternatively stated as the present value of all likely rating paths of the bond, discounted against the market risk-bearing bond forward rates of the different rating categories. The impact of term structure volatility and rating migration uncertainty on bond valuation is also considered.It is shown that the relationship between rating migration and default probability is complex, and the default probabilities of different rating categories are time-dependent and not isolated from each other. Also, rating migration resembles a delayed default process that influences default probabilities of subsequent intervals. The implications of a rating migration matrix may perhaps only be fully understood through simulation. This form one of the first points by which to evaluate rating migration matrices. The results of the valuation model show that historical rating migration matrices may not be optimal for pricing bonds ahistorically. A principal premise of the study is the dichotomy between historical values and ahistorical estimates, particularly with regards to rating migration. It is argued that historical estimates face two key shortcomings: they must be able to accurately forecast future rating migration and rating category intensities as a result, and they must specify a method to include rating migration uncertainty. An optimization model is delineated to extract ahistorical rating migration matrices from market prices. This too has implications that should be considered. In light of the above, reduced form models may have an advantage over structural models, in their ability to portray a far more sophisticated default process.

Rating Migration and Bond Valuation: Towards Ahistorical Rating Migration Matrices and Default Probability Term Structures

The study examines rating migration, and default probability term structures obtained from rating migration matrices. It expands on the use of rating migration matrices with reduced form bond valuation models, by formally delineating the probability of default according to the likely rating paths of a bond, as implied by the rating migration matrix. Further, two alternatives are also considered. First, the cost of default is stipulated as the recovery of par according to the exit rating upon default. Also, in addition to stating the value of a bond in terms of expected cash flows, when considering the probability of default, the value of a bond is alternatively stated as the present value of all likely rating paths of the bond, discounted against the market risk-bearing bond forward rates of the different rating categories. The impact of term structure volatility and rating migration uncertainty on bond valuation is also considered.It is shown that the relationship between rating migration and default probability is complex, and the default probabilities of different rating categories are time-dependent and not isolated from each other. Also, rating migration resembles a delayed default process that influences default probabilities of subsequent intervals. The implications of a rating migration matrix may perhaps only be fully understood through simulation. This form one of the first points by which to evaluate rating migration matrices. The results of the valuation model show that historical rating migration matrices may not be optimal for pricing bonds ahistorically. A principal premise of the study is the dichotomy between historical values and ahistorical estimates, particularly with regards to rating migration. It is argued that historical estimates face two key shortcomings: they must be able to accurately forecast future rating migration and rating category intensities as a result, and they must specify a method to include rating migration uncertainty. An optimization model is delineated to extract ahistorical rating migration matrices from market prices. This too has implications that should be considered. In light of the above, reduced form models may have an advantage over structural models, in their ability to portray a far more sophisticated default process.

Extraction of Implied Default Probabilities and Expected Recovery Values from a Combination of Bond Prices and Credit Default Swap Spreads

This paper proposes an intuitively simple technique of inferring a term structure of implied default probabilities and a term structure of expected recovery values from a combination of market bond prices and credit default swap spreads. Credit risk for longer maturities can not only be characterized by higher implied default probabilities but also be associated with downward revisions in the expected recovery value. Allowing for a non-flat recovery value term structure along with varying implied default probabilities across different periods can be a useful relaxation of the restriction of a constant recovery rate that is typical to the existing studies on credit risk modeling. We demonstrate that a term structure of recovery values can be separately identified along with the term structure of implied default probabilities through the information contained in the term structure of bond prices and the term structure of credit default swaps with corresponding maturities. Solutions to a set of numerical examples are provided that can be replicated independently. The combination of implied default probabilities and expected recovery values provides one with a set of valuable tools to value a range of credit claims for which historical default and recovery rates are not readily available.

Dependent defaults and losses with factor copula models

Abstract We present a class of flexible and tractable static factor models for the term structure of joint default probabilities, the factor copula models. These high-dimensional models remain parsimonious with paircopula constructions, and nest many standard models as special cases. The loss distribution of a portfolio of contingent claims can be exactly and efficiently computed when individual losses are discretely supported on a finite grid. Numerical examples study the key features affecting the loss distribution and multi-name credit derivatives prices. An empirical exercise illustrates the flexibility of our approach by fitting credit index tranche prices.

Rating Migration and Bond Valuation: Decomposing Rating Migration Matrices from Market Data Via Default Probability Term Structures

The study builds on previous research that decomposes rating category default probability term structures from rating category interest rate term structures, and proposes a method to decompose rating migration matrices from market data, via decomposed default probability term structures. To investigate the power and accuracy of the proposed method, it was examined to what extent an existing, known rating migration matrix could again be surfaced by the method. Overall, the results are more than satisfactory, and the method promises to be accurate. Although not considered here, the main objective is the application of the method to market data. The outcome should be insightful in itself, and can be used to evaluate historical rating migration matrices commonly devised by rating agencies, and to form a better understanding of the default probability term structures embedded in market data.

Rating Migration and Bond Valuation: Ahistorical Interest Rate and Default Probability Term Structures

The study extends the theoretical framework proposed to decompose rating migration matrices from bond market price data. In this context, method to decompose default probability term structures for and from interest rate term structures for different rating categories, is also delineated and empirically evaluated. In principle, whenever it is possible to decompose an interest rate term structure for a rating category from ahistorical market data, it should also be possible to decompose a default probability term structure for the rating category, and to decompose a default probability term structure for the particular interest rate term structure of the rating category. Principal to the method and vantage point is that, when decomposing a default probability term structure, purely ahistorical market data is used, as is the case when decomposing interest rate term structures. In addition, greater emphasis is placed on surfacing actual market perceptions regarding default probability, which may differ from theoretical modelling of default probability. The method naturally allows a mapping and transitioning between interest rate term structures and default probability term structures. Consequentially, the study examines the corresponding interest rate term structures of the default probability term structures of a typical rating migration matrix, and the corresponding default probability term structures of a typical market interest rate term structure set. The sensitivity of the results to the coupon rate used is also examined, particularly in view of using an external (artificial) portfolio to facilitate the process. It is found that the default probability term structures decomposed from market interest rate term structures significantly differ from rating migration matrix based default probability term structures. This may point to differing views on default probability term structures. In addition, it is also shown that even external (artificial) portfolios can illuminate the default probability term structures of interest rate term structures with reasonable accuracy. Mapping to and fro interest rate term structures and default probability term structures introduces and additional level of triangulation and evaluation. It is expected that interest rate term structures should have valid and sensible default probability term structures, and vice versa. Another important implication of decomposing default probability term structures from ahistorical market data is that the default probability term structures would change at the frequency of the market and market data. This implies that market based default probability term structures may include a volatility component. Market based ahistorical default probability term structures offer a different and unique perspective on default probability term structures. If found to be an accurate representation of market views and perceptions, it may demand a reconsideration of the prominence of default probability as a bond valuation factor.

Risk management of Credit Default Swap

Credit derivatives are an up to date innovation in financial markets. These financial instrument have a potential to allow enterprises to trade and manage the credit risks and market risks. The striking growth of credit derivatives suggest that participant of financial markets find them to be useful instrument for risk management. The most popular and fundamental credit derivatives is a credit default swaps (CDS). In the paper we detailed the risk management of the credit default swaps and quantified the credit risk of investors in two way: (i) calculate the term structure of default probabilities from the market prices of traded CDS and (ii) calculate prices of CDS from the probability distribution of the time-to-default  Keywords: credit risk; credit default swap; risk management

Risk management of Credit Default Swap

Credit derivatives are an up to date innovation in financial markets. These financial instrument have a potential to allow enterprises to trade and manage the credit risks and market risks. The striking growth of credit derivatives suggest that participant of financial markets find them to be useful instrument for risk management. The most popular and fundamental credit derivatives is a credit default swaps (CDS). In the paper we detailed the risk management of the credit default swaps and quantified the credit risk of investors in two way: (i) calculate the term structure of default probabilities from the market prices of traded CDS and (ii) calculate prices of CDS from the probability distribution of the time-to-default  Keywords: credit risk; credit default swap; risk management

Measuring Credit Risk of Individual Corporate Bonds in US Energy Sector

In this paper, using the measures of the credit risk price spread (CRiPS) and the standardized credit risk price spread (S-CRiPS) proposed in Kariya’s (A CB (corporate bond) pricing model for deriving default probabilities and recovery rates. Eaton, IMS Collection Series: Festschrift for Professor Morris L., 2013) corporate bond model, we make a comprehensive empirical credit risk analysis on individual corporate bonds (CBs) in the US energy sector, where cross-sectional CB and government bond price data is used with bond attributes. Applying the principal component analysis method to the S-CRiPSs, we also categorize individual CBs into three different groups and study on their characteristics of S-CRiPS fluctuations of each group in association with bond attributes. Secondly, using the market credit rating scheme proposed by Kariya et al. (2014), we make credit-homogeneous groups of CBs and show that our rating scheme is empirically very timely and useful. Thirdly, we derive term structures of default probabilities for each homogeneous group, which reflect the investors’ views and perspectives on the future default probabilities or likelihoods implicitly implied by the CB prices for each credit-homogeneous group. Throughout this paper it is observed that our credit risk models and the associated measures for individual CBs work effectively and can timely provide the market credit information evaluated by investors.

Dependent Defaults and Losses with Factor Copula Models

We introduce a class of flexible and tractable static factor models for the joint term structure of default probabilities, the factor copula models. These high dimensional models remain parsimonious with pair copula constructions, and nest numerous standard models as special cases. With finitely supported random losses, the loss distributions of credit portfolios and derivatives can be exactly and efficiently computed. Numerical examples on collateral debt obligation (CDO), CDO squared, and credit index swaption illustrate the versatility of our framework. An empirical exercise shows that a simple model specification can fit credit index tranche prices.

Multiperiod Default Premia for Private Firms

We estimate term structures of default probabilities for private firms using data consisting of 1,759 default events from a sample of 29,894 firms in the period between 1999 and 2014. Each firm's default likelihood is characterized by a forward intensity model employing both macro risk factors and firm-specific attributes derived from financial statements. As private firms do not have traded stock prices, we devise a methodology to obtain a public-firm equivalent distance-to-default by projection which references the distance-to-defaults of public firms with comparable firm attributes. The fitted model provides accurate multi-period forecasts of defaults for privately held firms. The reported interest rates charged to private firms are reflective of the estimated default term structure.

A dynamic program for valuing corporate securities

We design and implement a dynamic program for valuing corporate securities, seen as derivatives on a firm’s assets, and computing the term structure of yield spreads and default probabilities. Our setting is flexible for it accommodates an extended balance-sheet equality, arbitrary corporate debts, multiple seniority classes, and a reorganization process. This flexibility comes at the expense of a minor loss of efficiency. The analytical approach proposed in the literature is exchanged here for a quasi-analytical approach based on dynamic programming coupled with finite elements. To assess our construction, which shows flexibility and efficiency, we carry out a numerical investigation along with a complete sensitivity analysis.

Credit Risk Analysis on Euro Government Bonds-Term Structures of Default Probabilities

In this paper, we make a comprehensive credit risk analysis on government bonds (GBs) of Germany, France, Italy, Spain and Greece over the period 2007.4–2012.3, where interest rate (IR) differential, GB price differential, default probability (DP) and credit default swap (CDS) are considered. First, applying the GB-pricing model in Kariya (Quantitative methods for portfolio analysis: MTV approach. Springer, Berlin, 1993) to these GB prices, we derive the term structures of interest rates (TSIRs) and discuss on the Maastricht convergence condition for the IR-differentials among these states relative to the German TSIRs and make some observations on some divergent tendencies. The results are associated with the business cycles and budgetary condition of each state. In the second part, to substantiate this viewpoint, we first make credit risk price spread analysis on price differentials and derive the term structures of default probabilities (TSDPs) of the French, Italian, Spanish and Greek GBs relative to the German GBs, where the corporate bond (CB) model proposed in Kariya (Advances in modern statistical theory and applications: a Festschrift for Professor Morris L. Eaton. Institute of Mathematical Statistics, Beachwood, 2013) is used in the derivation. Then it is empirically shown that the TSDPs show a significant divergent movement at the end of 2011, affected by the Euro Crisis. In addition, the TSDPs of these GBs are empirically shown to be almost linear functions of the differences of the TSIRs, which enables us to state the Maastricht condition in terms of DP. Thirdly the effectiveness of our TSDPs is empirically verified by comparing them with the corresponding CDSs against US dollars.

Market-Based Credit Ratings

We present a methodology for rating in real-time the creditworthiness of public companies in the U.S. from the prices of traded assets. Our approach uses asset pricing data to impute a term structure of risk neutral survival functions or default probabilities. Firms are then clustered into ratings categories based on their survival functions using a functional clustering algorithm. This allows all public firms whose assets are traded to be directly rated by market participants. For firms whose assets are not traded, we show how they can be indirectly rated by matching them to firms that are traded based on observable characteristics. We also show how the resulting ratings can be used to construct loss distributions for portfolios of bonds. Finally, we compare our ratings to Standard & Poors and find that, over the period 2005 to 2011, our ratings lead theirs for firms that ultimately default.

Liquidity, leverage, and Lehman: A structural analysis of financial institutions in crisis

This paper presents a flexible, lattice-based structural credit risk model that uses equity market information and a detailed depiction of a financial institution’s liability structure to analyze default risk. The model is applied to examine the term structure of default probabilities for Lehman Brothers prior to its demise. The results indicate, as early as March, that the firm would likely lose access to external capital within two years. The model can be used as both a diagnostic tool for the early detection of financial distress and a prescriptive tool for addressing the sources of risk in large, complex financial institutions.

Market-Based Credit Ratings

We present a methodology for rating the credit worthiness of public companies in the U.S. from the prices of traded assets. Our approach uses asset pricing data to impute a term structure of risk neutral survival functions or default probabilities. Firms are then clustered into ratings categories based on their survival functions using a functional clustering algorithm. This allows all public firms whose assets are traded to be directly rated by market participants. For firms whose assets are not traded, we show how they can be indirectly rated through the use of matching estimators. We also show how the resulting ratings can be used to construct loss distributions for portfolios of bonds. Our approach has the advantages of being transparent, computationally tractable, simple to implement, and easy to interpret economically.

Liquidity, Leverage, and Lehman: A Structural Analysis of Financial Institutions in Crisis

The failure of Lehman Brothers highlighted the severe lapses in risk management and regulatory oversight that brought on and intensified the global financial crisis. This paper presents a structural credit risk model that provides useful early warning signals that regulators could have used to predict Lehman's future demise. Our easily implementable, flexible lattice-based model uses equity market information along with a complete depiction of Lehman's liability structure to analyze the term structure of the firm's default probabilities on a month-by-month basis throughout 2008. We find that the evolution of forward default probabilities indicates, as early as March, that the firm would likely lose access to external capital within two years. Our findings support regulators' suspicions that over-reliance on short-term funding and insufficient collateral compounded the effects of dangerously high leverage and resulted in undercapitalization and excessive risk exposure for Lehman Brothers. The model reinforces the importance of the relationship between capital structure and risk management. The contribution extends beyond the clinical analysis of Lehman Brothers' failure. The model can be used as both a diagnostic tool for the early detection of financial distress and a prescriptive tool for addressing the sources of risk in large, complex financial institutions.