March  2022, 18(2): 1321-1337. doi: 10.3934/jimo.2021022

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 Published  March 2022 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 and Management Optimization, 2022, 18 (2) : 1321-1337. doi: 10.3934/jimo.2021022
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. 

[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.

[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.

[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.

[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.

[6]

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

[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.

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

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

[10]

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

[11]

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

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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. 

[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.

[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.

[23]

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

[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.

[25]

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

[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.

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

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[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.

[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.

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. 

[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.

[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.

[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.

[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.

[6]

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

[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.

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

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

[10]

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

[11]

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

[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).

[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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[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. 

[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.

[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.

[23]

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

[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.

[25]

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

[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.

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

[29]

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

[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.

[31]

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

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[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.

[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.

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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