In both financial theory and practice, Value-at-risk (VaR) has become the predominant risk measure in the last two decades. Nevertheless, there is a lively and controverse on-going discussion about possible alternatives. Against this background, our first objective is to provide a current overview of related competitors with the focus on credit risk management which includes definition, references, striking properties and classification. The second part is dedicated to the measurement of risk concentrations of credit portfolios. Typically, credit portfolio models are used to calculate the overall risk (measure) of a portfolio. Subsequently, Euler's allocation scheme is applied to break the portfolio risk down to single counterparties (or different subportfolios) in order to identify risk concentrations. We first carry together the Euler formulae for the risk measures under consideration. In two cases (Median Shortfall and Range-VaR), explicit formulae are presented for the first time. Afterwards, we present a comprehensive study for a benchmark portfolio according to Duellmann and Masschelein (2007) and nine different risk measures in conjunction with the Euler allocation. It is empirically shown that-in principle-all risk measures are capable of identifying both sectoral and single-name concentration. However, both complexity of IT implementation and sensitivity of the risk figures w.r.t. changes of portfolio quality vary across the specific risk measures.
This article introduces the R package ctmcd, which provides an implementation of methods for the estimation of the parameters of a continuous-time Markov chain given that data are only available on a discrete-time basis. This data consists of partial observations of the state of the chain, which are made without error at discrete times, an issue also known as the embedding problem for Markov chains. The functions provided comprise matrix logarithm based approximations as described in Israel et al. (2001), as well as Kreinin and Sidelnikova (2001), an expectation-maximization algorithm and a Gibbs sampling approach, both introduced by Bladt and Sørensen (2005). For the expectationmaximization algorithm Wald confidence intervals based on the Fisher information estimation method of Oakes (1999) are provided. For the Gibbs sampling approach, equal-tailed credibility intervals can be obtained. In order to visualize the parameter estimates, a matrix plot function is provided. The methods described are illustrated by Standard and Poor's discrete-time corporate credit rating transition data.
We present two methodologies on the estimation of rating transition probabilities within Markov and non-Markov frameworks. We first estimate a continuous-time Markov chain using discrete (missing) data and derive a simpler expression for the Fisher information matrix, reducing the computational time needed for the Wald confidence interval by a factor of a half. We provide an efficient procedure for transferring such uncertainties from the generator matrix of the Markov chain to the corresponding rating migration probabilities and, crucially, default probabilities. For our second contribution, we assume access to the full (continuous) data set and propose a tractable and parsimonious self-exciting marked point processes model able to capture the non-Markovian effect of rating momentum. Compared to the Markov model, the non-Markov model yields higher probabilities of default in the investment grades, but also lower default probabilities in some speculative grades. Both findings agree with empirical observations and have clear practical implications. We use Moody's proprietary corporate credit rating data set. Parts of our implementation are available in the R package ctmcd.
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