A Markov Model for the Term Structure of Credit Risk Spreads

Jarrow, Robert A.; Lando, David; Turnbull, Stuart M.

doi:10.1093/rfs/10.2.481

Cited by 1,226 publications

(515 citation statements)

References 21 publications

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“…First, regularization techniques of adjusting negative off-diagonal entries, arising when computing the matrix logarithm, can be found in [3,4], where DA and WA are described. A similar procedure is given in the seminal work of [26]. These are further investigated for example in [1,8,14].…”

Section: Optimization Problemsmentioning

confidence: 99%

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Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

Hughes

Werner

2016

Risks

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Transition matrices, containing credit risk information in the form of ratings based on discrete observations, are published annually by rating agencies. A substantial issue arises, as for higher rating classes practically no defaults are observed yielding default probabilities of zero. This does not always reflect reality. To circumvent this shortcoming, estimation techniques in continuous-time can be applied. However, raw default data may not be available at all or not in the desired granularity, leaving the practitioner to rely on given one-year transition matrices. Then, it becomes necessary to transform the one-year transition matrix to a generator matrix. This is known as the embedding problem and can be formulated as a nonlinear optimization problem, minimizing the distance between the exponential of a potential generator matrix and the annual transition matrix. So far, in credit risk-related literature, solving this problem directly has been avoided, but approximations have been preferred instead. In this paper, we show that this problem can be solved numerically with sufficient accuracy, thus rendering approximations unnecessary. Our direct approach via nonlinear optimization allows one to consider further credit risk-relevant constraints. We demonstrate that it is thus possible to choose a proper generator matrix with additional structural properties.

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Section: Optimization Problemsmentioning

confidence: 99%

“…With decreasing ratings, the risk increases. Following [26] (also found in [14]; the corresponding proof is given in [27]), this requirement can be ensured by one of the following two conditions of Condition 3.…”

Section: Definition 4 (Migration Rate Constraints)mentioning

confidence: 99%

Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

Hughes

Werner

2016

Risks

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“…transition matrices captures this skewed behaviour better than methods using average transition matrices and adjusting them to the observed transition rates, as in Jarrow, Lando, and Turnbull (1997). We extract yearly transition matrices from Moody's annual default studies.…”

Section: Actual Default Intensities: Transition Matricesmentioning

confidence: 99%

The Relationship between Risk-Neutral and Actual Default Probabilities: The Credit Risk Premium

et al. 2015

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The study investigates empirically the relationship between the risk-neutral measure Q and the real-world measure P. We study the ratio between the risk-neutral and actual default intensities, which we call the coverage ratio or the relative credit risk premium. Actual default intensities are derived from rating agencies annual transition matrices, while risk-neutral default intensities are bootstrapped from CDS quotes of European corporates. We quantify the average risk premium and its changes over time. Compared to related literature, special attention is given to the effects of the recent financial and European sovereign crises. We find that average credit risk premia rose substantially and that post-crisis levels are still higher than those observed before the financial crisis. This observation is especially true for high-quality debt and if it persists, it will have an impact on corporates funding costs. The quantification and revision of risk premia contributes to the discussion of the credit spread puzzle and could give extra insights in valuation models that start from real-world estimates. Our work is furthermore important in the context of state aid assessment. The real economic value (REV) methodology, applied by the European Commission to evaluate impaired portfolios, is based on a long-term average risk premium.

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“…The determination of stochastic roots of stochastic matrices has found many applications in different areas of applied mathematics [9][10][11][12]. For instance, in economical applications credit ratings for a company are represented by a stochastic matrix recording the probability that the company changes from a credit rating to another [10,11].…”

Section: On Roots Of Stochastic Matricesmentioning

confidence: 99%

“…For instance, in economical applications credit ratings for a company are represented by a stochastic matrix recording the probability that the company changes from a credit rating to another [10,11]. These transition matrices are recorded for a given time interval, which usually is one year.…”

Section: On Roots Of Stochastic Matricesmentioning

confidence: 99%

Information mobility in complex networks

Estrada

2009

Phys. Rev. E

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The concept of information mobility in complex networks is introduced on the basis of a stochastic process taking place in the network. The transition matrix for this process represents the probability that the information arising at a given node is transferred to a target one. We use the fractional powers of this transition matrix to investigate the stochastic process at fractional time intervals. The mobility coefficient is then introduced on the basis of the trace of these fractional powers of the stochastic matrix. The fractional time at which a network diffuses 50% of the information contained in its nodes ( 50 / 1 k ) is also introduced. We then show that the scale-free random networks display better spread of information than the non scale-free ones. We study 38 real-world networks and analyze their performance in spreading information from their nodes. We find that some real-world networks perform even better than the scale-free networks with the same average degree and we point out some of the structural parameters that make this possible.

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A Markov Model for the Term Structure of Credit Risk Spreads

Cited by 1,226 publications

References 21 publications

Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

Choosing Markovian Credit Migration Matrices by Nonlinear Optimization

The Relationship between Risk-Neutral and Actual Default Probabilities: The Credit Risk Premium

Information mobility in complex networks

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