Estimation of risk contributions with MCMC

Abstract

Determining risk contributions of unit exposures to portfolio-wide economic capital is an important task in financial risk management. Computing risk contributions involves difficulties caused by rare-event simulations. In this study, we address the problem of estimating risk contributions when the total risk is measured by value-at-risk (VaR). Our proposed estimator of VaR contributions is based on the Metropolis-Hasting (MH) algorithm, which is one of the most prevalent Markov chain Monte Carlo (MCMC) methods. Unlike existing estimators, our MH-based estimator consists of samples from the conditional loss distribution given a rare event of interest. This feature enhances sample efficiency compared with the crude Monte Carlo method. Moreover, our method has consistency and asymptotic normality, and is widely applicable to various risk models having a joint loss density. Our numerical experiments based on simulation and real-world data demonstrate that in various risk models, even those having high-dimensional (≈500) inhomogeneous margins, our MH estimator has smaller bias and mean squared error when compared with existing estimators.

Keywords:

• Value-at-risk
• Risk allocation
• Risk contributions
• VaR contributions
• Copulas
• Markov chain Monte Carlo
• Metropolis-Hastings algorithm

JEL Classification:

• C58
• C63

Acknowledgments

We wish to thank to Paul Embrechts from ETH Zürich for his valuable comments regarding the simulation setup. We would also like to express our gratitude to Kengo Kamatani from Osaka University, and Marius Hofert from the University of Waterloo for fruitful discussions on MCMC and Archimedean copulas. Finally, we are thankful to an associate editor and anonymous referees for their careful reading of the manuscript and their insightful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Supplemental data

Supplemental data for this article can be accessed at https://doi.org/14697688.2019.1588469.

ORCID

Takaaki Koike http://orcid.org/0000-0002-7940-1418

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science (JSPS) under the Core-to-Core program at Keio University.