Operational risk quantified with spectral risk measures: a refined closed-form approximation

Abstract

The quantification of operational risk has become an important issue as a result of the new capital charges required by the Basel Capital Accord (Basel II) to cover the potential losses of this type of risk. In this paper, we investigate second-order approximation of operational risk quantified with spectral risk measures (OpSRMs) within the theory of second-order regular variation (2RV) and second-order subexponentiality. The result shows that asymptotically two cases (the fast convergence case and the slow convergence) arise depending on the range of the second-order parameter. We also show that the second-order approximation under 2RV is asymptotically equivalent to the slow convergence case. A number of Monte Carlo simulations for a range of empirically relevant frequency and severity distributions are employed to illustrate the performance of our second-order results. The simulation results indicate that our second-order approximations tend to reduce the estimation errors to a great degree, especially for the fast convergence case, and are able to capture the sub-extremal behavior of OpSRMs better than the first-order approximation. Our asymptotic results have implications for the regulation of financial institutions, and may provide further insights into the measurement and management of operational risk.

Keywords:

• Operational risk
• Spectral risk measures
• Regular variation
• Second-order regular variation
• Asymptotically smooth
• Second-order subexponentiality

JEL Classification:

• C14
• C16
• C65
• G21

Acknowledgments

The authors would like to thank two anonymous referees for careful reading of the paper and for helpful comments that greatly improved the paper. We are grateful to Prof. Edward Omey for sending us a copy of his early research papers on the tail behavior of subexponential distributions. We would like to thank seminar participants at the tenth International Symposium on Financial System Engineering and Risk Management (FSERM, 2012) in China for helpful comments and suggestions. All remaining errors are ours alone.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Xundi Diao http://orcid.org/0000-0002-7691-7208

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [grant numbers 71790592, 71771147 and 71320107002].