doi: 10.3934/jimo.2021022
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Asymptotics for VaR and CTE of total aggregate losses in a bivariate operational risk cell model

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

Received  July 2020 Revised  October 2020 Early access January 2021

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.

Citation: Yishan Gong, Yang Yang. Asymptotics for VaR and CTE of total aggregate losses in a bivariate operational risk cell model. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021022
References:
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C. AngelaR. BisignaniG. Masala and M. Micocci, Advanced operational risk modelling in banks and insurance companies, Investment Management and Financial Innovations, 6 (2009), 78-83.   Google Scholar

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A. V. Asimit and A. L. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model, Scandinavian Actuarial Journal, (2010), 93–104. doi: 10.1080/03461230802700897.  Google Scholar

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A. V. AsimitE. FurmanQ. Tang and R. Vernic, Asymptotics for risk capital allocations based on conditional tail expectation, Insurance: Mathematics and Economics, 49 (2011), 310-324.  doi: 10.1016/j.insmatheco.2011.05.002.  Google Scholar

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A. L. BadescuG. LanX. S. Lin and D. Tang, Modeling correlated frequencies with application in operational risk management, Journal of Operational Risk, 10 (2015), 1-43.  doi: 10.21314/JOP.2015.157.  Google Scholar

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S. A. BakarN. A. HamzahM. Maghsoudi and S. Nadarajah, Modeling loss data using composite models, Insurance: Mathematics and Economics, 61 (2015), 146-154.  doi: 10.1016/j.insmatheco.2014.08.008.  Google Scholar

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Basel Committee on Banking Supervision, International Convergence of Capital Measurement and Capital Standards, Report of Basel Committee on Banking Supervision, 2004. Google Scholar

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Basel Committee on Banking Supervision, Consultative Document, Fundamental Review of the Trading Book: A revised Market Risk framework, Report of Basel Committee on Banking Supervision, 2013. Google Scholar

[8] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge university press, Cambridge, 1989.   Google Scholar
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K. Böcker and C. Klüppelberg, Operational VaR: A closed-form approximation, Risk, 12 (2005), 90-93.   Google Scholar

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K. Böcker and C. Klüppelberg, Multivariate models for operational risk, Quantitative Finance, 10 (2010), 855-869.  doi: 10.1080/14697680903358222.  Google Scholar

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E. BrechmannC. Czado and S. Paterlini, Modeling dependence of operational loss frequencies, Journal of Operational Risk, 8 (2013), 105-126.   Google Scholar

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N. Cantle, D. Clark, J. Kent and H. Verheugen, A brief overview of current approaches to operational risk under Solvency II, Milliman White Paper, (2012). Google Scholar

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V. Chavez-DemoulinP. Embrechts and J. Nešlehová, Quantitative models for operational risk: Extremes, dependence and aggregation, Journal of Banking & Finance, 30 (2006), 2635-2658.  doi: 10.1016/j.jbankfin.2005.11.008.  Google Scholar

[14]

Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stochastic Models, 25 (2009), 76-89.  doi: 10.1080/15326340802641006.  Google Scholar

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K. Coorey and M. M. Ananda, Modelling actuarial data with a composite log-normal-Pareto model, Scandinavian Actuarial Journal, (2005), 321–334. doi: 10.1080/03461230510009763.  Google Scholar

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P. de Fontnouvelle, D. Jesus-Rueff, J. S. Jordan and E. S. Rosengren, Using Loss Data to Quantify Operational Risk, Working Paper, 2003. doi: 10.2139/ssrn.395083.  Google Scholar

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M. Eling, Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?, Insurance: Mathematics and Economics, 51 (2012), 239-248.  doi: 10.1016/j.insmatheco.2012.04.001.  Google Scholar

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P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997. Google Scholar

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S. EmmerM. Kratz and D. Tasche, What is the best risk measure in practice? A comparison of standard measures, Journal of Risk, 18 (2015), 31-60.   Google Scholar

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S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions, Springer, New York, 2011. doi: 10.1007/978-1-4419-9473-8.  Google Scholar

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J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, Journal of Theoretical Probability, 22 (2009), 871-882.  doi: 10.1007/s10959-008-0159-5.  Google Scholar

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P. HartmannS. Straetmans and C. G. de Vries, Heavy tails and currency crises, Journal of Empirical Finance, 17 (2010), 241-254.   Google Scholar

[24]

H. Joe and H. Li, Tail risk of multivariate regular variation, Methodology and Computing in Applied Probability, 13 (2011), 671-693.  doi: 10.1007/s11009-010-9183-x.  Google Scholar

[25]

E. L. Lehmann, Some concepts of dependence, The Annals of Mathematical Statistics, 37 (1966), 1137-1153.  doi: 10.1214/aoms/1177699260.  Google Scholar

[26]

J. LiQ. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Advances in Applied Probability, 42 (2010), 1126-1146.  doi: 10.1239/aap/1293113154.  Google Scholar

[27] A. J. McNeilR. Frey and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, Princeton, 2015.   Google Scholar
[28]

M. Moscadelli, The Modelling of Operational Risk: Experience with the Analysis of the Data Collected by the Basel Committee, Technical report by Bank of Italy, 2004. doi: 10.2139/ssrn.557214.  Google Scholar

[29]

S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer, New York, 1987.  Google Scholar

[30]

S. I. Resnick, Discussion of the Danish data on large fire insurance losses, ASTIN Bulletin, 27 (1997), 139-151.  doi: 10.2143/AST.27.1.563211.  Google Scholar

[31]

S. I. Resnick, Heavy-tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York, 2007.  Google Scholar

[32]

Q. Tang, The finite-time ruin probability of the compound Poisson model with constant interest force, Journal of Applied Probability, 42 (2005), 608-619.  doi: 10.1239/jap/1127322015.  Google Scholar

[33]

Q. TangZ. Tang and Y. Yang, Sharp asymptotics for large portfolio losses under extreme risks, European Journal of Operational Research, 276 (2019), 710-722.  doi: 10.1016/j.ejor.2019.01.025.  Google Scholar

[34]

Q. Tang and Y. Yang, Interplay of insurance and financial risks in a stochastic environment, Scandinavian Actuarial Journal, (2019), 432–451. doi: 10.1080/03461238.2019.1573753.  Google Scholar

[35]

Q. Tang and Z. Yuan, Asymptotic analysis of the loss given default in the presence of multivariate regular variation, North American Actuarial Journal, 17 (2013), 253-271.  doi: 10.1080/10920277.2013.830557.  Google Scholar

[36]

Y. Yang, Y. Gong and J. Liu, Measuring tail operational risk in univariate and multivariate models under extreme losses, Working Paper, (2019). Google Scholar

[37]

Y. Yang, T. Jiang, K. Wang and K. C. Yuen, Interplay of financial and insurance risks in dependent discrete-time risk models, Statistics & Probability Letters, 162 (2020), 108752, 11 pp. doi: 10.1016/j.spl.2020.108752.  Google Scholar

[38]

Y. YangK. WangJ. Liu and Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, Journal of Industrial & Management Optimization, 15 (2019), 481-505.  doi: 10.3934/jimo.2018053.  Google Scholar

[39]

Y. YangK. C. Yuen and J. Liu, Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims, Journal of Industrial & Management Optimization, 14 (2018), 231-247.  doi: 10.3934/jimo.2017044.  Google Scholar

[40]

X. ZhuY. Wang and J. Li, Operational risk measurement: A loss distribution approach with segmented dependence, Journal of Operational Risk, 14 (2019), 1-20.  doi: 10.21314/JOP.2019.220.  Google Scholar

show all references

References:
[1]

C. AngelaR. BisignaniG. Masala and M. Micocci, Advanced operational risk modelling in banks and insurance companies, Investment Management and Financial Innovations, 6 (2009), 78-83.   Google Scholar

[2]

A. V. Asimit and A. L. Badescu, Extremes on the discounted aggregate claims in a time dependent risk model, Scandinavian Actuarial Journal, (2010), 93–104. doi: 10.1080/03461230802700897.  Google Scholar

[3]

A. V. AsimitE. FurmanQ. Tang and R. Vernic, Asymptotics for risk capital allocations based on conditional tail expectation, Insurance: Mathematics and Economics, 49 (2011), 310-324.  doi: 10.1016/j.insmatheco.2011.05.002.  Google Scholar

[4]

A. L. BadescuG. LanX. S. Lin and D. Tang, Modeling correlated frequencies with application in operational risk management, Journal of Operational Risk, 10 (2015), 1-43.  doi: 10.21314/JOP.2015.157.  Google Scholar

[5]

S. A. BakarN. A. HamzahM. Maghsoudi and S. Nadarajah, Modeling loss data using composite models, Insurance: Mathematics and Economics, 61 (2015), 146-154.  doi: 10.1016/j.insmatheco.2014.08.008.  Google Scholar

[6]

Basel Committee on Banking Supervision, International Convergence of Capital Measurement and Capital Standards, Report of Basel Committee on Banking Supervision, 2004. Google Scholar

[7]

Basel Committee on Banking Supervision, Consultative Document, Fundamental Review of the Trading Book: A revised Market Risk framework, Report of Basel Committee on Banking Supervision, 2013. Google Scholar

[8] N. H. BinghamC. M. Goldie and J. L. Teugels, Regular Variation, Cambridge university press, Cambridge, 1989.   Google Scholar
[9]

K. Böcker and C. Klüppelberg, Operational VaR: A closed-form approximation, Risk, 12 (2005), 90-93.   Google Scholar

[10]

K. Böcker and C. Klüppelberg, Multivariate models for operational risk, Quantitative Finance, 10 (2010), 855-869.  doi: 10.1080/14697680903358222.  Google Scholar

[11]

E. BrechmannC. Czado and S. Paterlini, Modeling dependence of operational loss frequencies, Journal of Operational Risk, 8 (2013), 105-126.   Google Scholar

[12]

N. Cantle, D. Clark, J. Kent and H. Verheugen, A brief overview of current approaches to operational risk under Solvency II, Milliman White Paper, (2012). Google Scholar

[13]

V. Chavez-DemoulinP. Embrechts and J. Nešlehová, Quantitative models for operational risk: Extremes, dependence and aggregation, Journal of Banking & Finance, 30 (2006), 2635-2658.  doi: 10.1016/j.jbankfin.2005.11.008.  Google Scholar

[14]

Y. Chen and K. C. Yuen, Sums of pairwise quasi-asymptotically independent random variables with consistent variation, Stochastic Models, 25 (2009), 76-89.  doi: 10.1080/15326340802641006.  Google Scholar

[15]

K. Coorey and M. M. Ananda, Modelling actuarial data with a composite log-normal-Pareto model, Scandinavian Actuarial Journal, (2005), 321–334. doi: 10.1080/03461230510009763.  Google Scholar

[16]

P. de Fontnouvelle, D. Jesus-Rueff, J. S. Jordan and E. S. Rosengren, Using Loss Data to Quantify Operational Risk, Working Paper, 2003. doi: 10.2139/ssrn.395083.  Google Scholar

[17]

L. de Haan and S. I. Resnick, On the observation closest to the origin, Stochastic Processes and their Applications, 11 (1981), 301-308.  doi: 10.1016/0304-4149(81)90032-6.  Google Scholar

[18]

M. Eling, Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?, Insurance: Mathematics and Economics, 51 (2012), 239-248.  doi: 10.1016/j.insmatheco.2012.04.001.  Google Scholar

[19]

P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events for Insurance and Finance, Springer, Berlin, 1997. Google Scholar

[20]

S. EmmerM. Kratz and D. Tasche, What is the best risk measure in practice? A comparison of standard measures, Journal of Risk, 18 (2015), 31-60.   Google Scholar

[21]

S. Foss, D. Korshunov and S. Zachary, An Introduction to Heavy-tailed and Subexponential Distributions, Springer, New York, 2011. doi: 10.1007/978-1-4419-9473-8.  Google Scholar

[22]

J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, Journal of Theoretical Probability, 22 (2009), 871-882.  doi: 10.1007/s10959-008-0159-5.  Google Scholar

[23]

P. HartmannS. Straetmans and C. G. de Vries, Heavy tails and currency crises, Journal of Empirical Finance, 17 (2010), 241-254.   Google Scholar

[24]

H. Joe and H. Li, Tail risk of multivariate regular variation, Methodology and Computing in Applied Probability, 13 (2011), 671-693.  doi: 10.1007/s11009-010-9183-x.  Google Scholar

[25]

E. L. Lehmann, Some concepts of dependence, The Annals of Mathematical Statistics, 37 (1966), 1137-1153.  doi: 10.1214/aoms/1177699260.  Google Scholar

[26]

J. LiQ. Tang and R. Wu, Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model, Advances in Applied Probability, 42 (2010), 1126-1146.  doi: 10.1239/aap/1293113154.  Google Scholar

[27] A. J. McNeilR. Frey and P. Embrechts, Quantitative Risk Management: Concepts, Techniques and Tools, Princeton University Press, Princeton, 2015.   Google Scholar
[28]

M. Moscadelli, The Modelling of Operational Risk: Experience with the Analysis of the Data Collected by the Basel Committee, Technical report by Bank of Italy, 2004. doi: 10.2139/ssrn.557214.  Google Scholar

[29]

S. I. Resnick, Extreme Values, Regular Variation and Point Processes, Springer, New York, 1987.  Google Scholar

[30]

S. I. Resnick, Discussion of the Danish data on large fire insurance losses, ASTIN Bulletin, 27 (1997), 139-151.  doi: 10.2143/AST.27.1.563211.  Google Scholar

[31]

S. I. Resnick, Heavy-tail Phenomena: Probabilistic and Statistical Modeling, Springer, New York, 2007.  Google Scholar

[32]

Q. Tang, The finite-time ruin probability of the compound Poisson model with constant interest force, Journal of Applied Probability, 42 (2005), 608-619.  doi: 10.1239/jap/1127322015.  Google Scholar

[33]

Q. TangZ. Tang and Y. Yang, Sharp asymptotics for large portfolio losses under extreme risks, European Journal of Operational Research, 276 (2019), 710-722.  doi: 10.1016/j.ejor.2019.01.025.  Google Scholar

[34]

Q. Tang and Y. Yang, Interplay of insurance and financial risks in a stochastic environment, Scandinavian Actuarial Journal, (2019), 432–451. doi: 10.1080/03461238.2019.1573753.  Google Scholar

[35]

Q. Tang and Z. Yuan, Asymptotic analysis of the loss given default in the presence of multivariate regular variation, North American Actuarial Journal, 17 (2013), 253-271.  doi: 10.1080/10920277.2013.830557.  Google Scholar

[36]

Y. Yang, Y. Gong and J. Liu, Measuring tail operational risk in univariate and multivariate models under extreme losses, Working Paper, (2019). Google Scholar

[37]

Y. Yang, T. Jiang, K. Wang and K. C. Yuen, Interplay of financial and insurance risks in dependent discrete-time risk models, Statistics & Probability Letters, 162 (2020), 108752, 11 pp. doi: 10.1016/j.spl.2020.108752.  Google Scholar

[38]

Y. YangK. WangJ. Liu and Z. Zhang, Asymptotics for a bidimensional risk model with two geometric Lévy price processes, Journal of Industrial & Management Optimization, 15 (2019), 481-505.  doi: 10.3934/jimo.2018053.  Google Scholar

[39]

Y. YangK. C. Yuen and J. Liu, Asymptotics for ruin probabilities in Lévy-driven risk models with heavy-tailed claims, Journal of Industrial & Management Optimization, 14 (2018), 231-247.  doi: 10.3934/jimo.2017044.  Google Scholar

[40]

X. ZhuY. Wang and J. Li, Operational risk measurement: A loss distribution approach with segmented dependence, Journal of Operational Risk, 14 (2019), 1-20.  doi: 10.21314/JOP.2019.220.  Google Scholar

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
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
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
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
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
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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