Abstract
We review recent progress in modeling credit risk for correlated assets. We employ a new interpretation of the Wishart model for random correlation matrices to model non-stationary effects. We then use the Merton model in which default events and losses are derived from the asset values at maturity. To estimate the time development of the asset values, the stock prices are used, the correlations of which have a strong impact on the loss distribution, particularly on its tails. These correlations are non-stationary, which also influences the tails. We account for the asset fluctuations by averaging over an ensemble of random matrices that models the truly existing set of measured correlation matrices. As a most welcome side effect, this approach drastically reduces the parameter dependence of the loss distribution, allowing us to obtain very explicit results, which show quantitatively that the heavy tails prevail over diversification benefits even for small correlations. We calibrate our random matrix model with market data and show how it is capable of grasping different market situations. Furthermore, we present numerical simulations for concurrent portfolio risks, i.e., for the joint probability densities of losses for two portfolios. For the convenience of the reader, we give an introduction to the Wishart random matrix model.
Highlights
To assess the impact of credit risk on the systemic stability of the financial markets and the economy as a whole is of considerable importance as the subprime crisis of 2007–2009 and the events following the collapse of Lehman Brothers drastically demonstrated (Hull 2009).Better credit risk estimation is urgently called for
We report the results of Monte Carlo simulations for the general case of empirical correlation matrices that yield the value at risk (VaR) and expected tail loss (ETL)
We review our own work on how to take into account the non-stationarity of asset correlations into credit risk (Schmitt et al 2013, 2014, 2015; Sicking et al 2018)
Summary
To assess the impact of credit risk on the systemic stability of the financial markets and the economy as a whole is of considerable importance as the subprime crisis of 2007–2009 and the events following the collapse of Lehman Brothers drastically demonstrated (Hull 2009). The problem to be addressed becomes a statistical one, as loss distributions for large portfolios of credit contracts have to be estimated They have a very heavy right tail, which is due to either unusually large single events such as the Enron bankruptcy or the simultaneous occurrence of many small events as seen during the subprime crisis. We report the results of Monte Carlo simulations for the general case of empirical correlation matrices that yield the value at risk (VaR) and expected tail loss (ETL) Another important aspect is comprised of concurrent losses of different portfolios. This review paper is organized as follows: In Section 2, we introduce random matrix theory for non-stationary asset correlations, including a sketch of the Wishart model for readers not familiar with random matrices
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