Asymptotics for VaR and CTE of total aggregate losses in a bivariate operational risk cell model

Yishan Gong; Yang Yang

doi:10.3934/jimo.2021022

Article Contents

2022, Volume 18, Issue 2: 1321-1337. Doi: 10.3934/jimo.2021022

This issue Previous Article On the optimal control problems with characteristic time control constraints Next Article Integrated dynamic interval data envelopment analysis in the presence of integer and negative data

Asymptotics for VaR and CTE of total aggregate losses in a bivariate operational risk cell model

Yishan Gong^1, and
Yang Yang^2, ,

1.
Department of Mathematical Sciences, Xi'an Jiaotong Liverpool University, Suzhou, Jiangsu 215123, China
2.
School of Mathematics and Statistics, Nanjing Audit University, Nanjing, Jiangsu 211815, China

^* Corresponding author: Yang Yang
^* Corresponding author: Yang Yang

Received: June 30, 2020

Revised: September 30, 2020

Early access: January 2021

Published: March 2022

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Abstract

This paper considers a bivariate operational risk cell model, in which the loss severities are modelled by some heavy-tailed and weakly (or strongly) dependent nonnegative random variables, and the frequency processes are described by two arbitrarily dependent general counting processes. In such a model, we establish some asymptotic formulas for the Value-at-Risk and Conditional Tail Expectation of the total aggregate loss. Some simulation studies are also conducted to check the accuracy of the obtained theoretical results via the Monte Carlo method.

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Mathematics Subject Classification: Primary: 62P05; Secondary: 62E20, 91B30.

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Figure 1. Comparison of the simulated and asymptotic estimates for $ \mathrm{VaR}_q(S(t)) $, with Frank dependent and Weibull distributed severities and AI or AD dependent frequency processes in Theorem 3.1

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Figure 2. Comparison of the simulated and asymptotic estimates for $ \mathrm{VaR}_q(S(t)) $, with Frank dependent and Pareto distributed severities and AI or AD dependent frequency processes in Theorem 3.2

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Figure 3. Comparison of the simulated and asymptotic estimates for $ \mathrm{CTE}_q(S(t)) $, with Frank dependent and Pareto distributed severities and AI or AD dependent frequency processes in Theorem 3.2

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Figure 4. Comparison of the simulated and asymptotic estimates for $ \mathrm{VaR}_q(S(t)) $, with Gumbel dependent and Pareto distributed severities and AI or AD dependent frequency processes in Theorem 3.3

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Figure 5. Comparison of the simulated and asymptotic estimates for $ \mathrm{CTE}_q(S(t)) $, with Gumbel dependent and Pareto distributed severities and AI or AD dependent frequency processes in Theorem 3.3

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Figure 6. Comparison of the simulated and asymptotic estimates for $ \mathrm{VaR}_q(S(t)) $, with common frequency process, and Gumbel or Frank dependent Pareto distributed severities in Theorems 3.2 and 3.3

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