Value at Risk dan Tail Value at Risk dari Peubah Acak Besarnya Kerugian yang Menyebar Alpha Power Pareto

Ruhiyat Ruhiyat, Berlian Setiawaty, Muwafiqo Zamzami Dhuha


Value at Risk (VaR) and Tail Value at Risk (TVaR) are two measures that are commonly used to quantify the risk associated with a loss severity distribution. In this paper, both values are calculated analytically and estimated using a Monte Carlo simulation when the loss severity random variable has an alpha power Pareto distribution. This distribution is the result of alpha power transformation on a Pareto distribution. The random numbers used in the Monte Carlo simulation are generated from the alpha power Pareto distribution using the inverse transformation technique. In the special case used, the estimated VaR and TVaR values obtained from the Monte Carlo simulation for some security levels used are close to the actual VaR and TVaR values as long as the number of random numbers generated in the Monte Carlo simulation is sufficiently large.


Alpha Power Pareto Distribution; Monte Carlo Simulation; Tail Value at Risk; Value at Risk

Full Text:



P. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed. New York: The McGraw-Hill Companies, Inc., 2007.

S. A. Klugman, H. H. Panjer, and G. E. Willmot, Loss Models: From Data to Decisions, 5th ed. New Jersey: John Wiley and Sons, Inc., 2019.

P. Artzner, F. Delbaen, J.-M. Eber, and D. Heath, “Coherent Measures of Risk,” Mathematical Finance, vol. 9, no. 3, pp. 203–228, jul 1999, doi: 10.1111/1467-9965.00068.

Y. He, L. Peng, D. Zhang, and Z. Zhao, “Risk Analysis via Generalized Pareto Distributions,” Journal of Business & Economic Statistics, vol. 40, no. 2, pp. 852–867, apr 2022, doi:10.1080/07350015.2021.1874390.

O. L. Gebizlioglu, B. S¸enoglu, and Y. M. Kantar, “Comparison of certain value-at-risk estimation methods for the two-parameter Weibull loss distribution,” Journal of Computational and Applied Mathematics, vol. 235, no. 11, pp. 3304–3314, apr 2011, doi:10.1016/

R. Rockafellar and S. Uryasev, “Conditional value-at-risk for general loss distributions,” Journal of Banking & Finance, vol. 26, no. 7, pp. 1443–1471, jul 2002, doi: 10.1016/S0378-4266(02)00271-6.

S. Vanduffel, X. Chen, J. Dhaene, M. Goovaerts, L. Henrard, and R. Kaas, “Optimal approximations for risk measures of sums of lognormals based on conditional expectations,” Journal of Computational and Applied Mathematics, vol. 221, no. 1, pp. 202–218, nov 2008, doi:10.1016/

D. C. Nath and J. Das, “Modeling of claim severity through the mixture of exponential distribution and computation of its probability of ultimate ruin,” Thailand Statistician, vol. 15, no. 2, pp. 128–148, 2017.

A. Moumeesri and T. Pongsart, “Bonus-Malus Premiums Based on Claim Frequency and the Size of Claims,” Risks, vol. 10, no. 9, pp. 181–202, sep 2022, doi: 10.3390/risks10090181.

N. E. Frangos and S. D. Vrontos, “Design of Optimal Bonus-Malus Systems With a Frequency and a Severity Component On an Individual Basis in Automobile Insurance,” ASTIN Bulletin, vol. 31, no. 1, pp. 1–22, may 2001, doi: 10.2143/AST.31.1.991.

Y. Wang and I. Hobæk Haff, “Focussed selection of the claim severity distribution,” Scandinavian Actuarial Journal, vol. 2019, no. 2, pp. 129–142, feb 2019, doi: 10.1080/03461238.2018.1519847.

S. Ihtisham, A. Khalil, S. Manzoor, S. A. Khan, and A. Ali, “Alpha-Power Pareto distribution: Its properties and applications,” PLOS ONE, vol. 14, no. 6, p. e0218027, jun 2019, doi:10.1371/journal.pone.0218027.

A. Mahdavi and D. Kundu, “A new method for generating distributions with an application to exponential distribution,” Communications in Statistics - Theory and Methods, vol. 46, no. 13,

pp. 6543–6557, jul 2017, doi: 10.1080/03610926.2015.1130839.

M. Nassar, A. Alzaatreh, M. Mead, and O. Abo-Kasem, “Alpha power Weibull distribution: Properties and applications,” Communications in Statistics - Theory and Methods, vol. 46, no. 20, pp. 10 236–10 252, oct 2017, doi:10.1080/03610926.2016.1231816.

S. W. Philbrick, “A practical guide to the single parameter Pareto distribution,” in Proceedings of the Casualty Actuarial Society, vol. LXXII, 1985, pp. 44–84.

A. M. Law, Simulation Modeling and Analysis, 5th ed. New York: McGraw-Hill Education, 2015.

D. B. Thomas and W. Luk, “Estimation of sample mean and variance for Monte-Carlo simulations,” in 2008 International Conference on Field-Programmable Technology. IEEE, dec 2008, pp. 89–96, doi: 10.1109/FPT.2008.4762370.

K. M. Ramachandran and C. P. Tsokos, Mathematical Statistics with Applications in R, 2nd ed. London: Elsevier, 2015, doi:10.1016/C2012-0-07341-3.

M. Bee, R. Benedetti, and G. Espa, “On maximum likelihood estimation of a Pareto mixture,” Computational Statistics, vol. 28, no. 1, pp. 161–178, feb 2013, doi: 10.1007/s00180-011-0291-z.

A. B. Kisabo, N. C. Uchenna, and F. A. Adebimpe, “Newton’s Method for Solving Non-Linear System of Algebraic Equations (NLSAEs) with MATLAB/Simulink R and MAPLE R ,” vol. 2, no. 4, pp. 117–131, 2017, doi: 10.11648/j.ajmcm.20170204.14.

Sheldon Ross, Simulation, 6th ed. Cambridge, Massachusetts: Elsevier, 2022.

M. Snipes and D. C. Taylor, “Model selection and Akaike Information Criteria: An example from wine ratings and prices,” Wine Economics and Policy, vol. 3, no. 1, pp. 3–9, jun 2014, doi:10.1016/j.wep.2014.03.001.

M. E. Emetere, Numerical Methods in Environmental Data Analysis. Amsterdam: Elsevier, 2022, doi: 10.1016/C2018-0-04933-6.

R. V. Hogg and S. A. Klugman, “On the estimation of long tailed skewed distributions with actuarial applications,” Journal of Econometrics, vol. 23, no. 1, pp. 91–102, sep 1983, doi:10.1016/0304-4076(83)90077-5.


Copyright (c) 2023 Ruhiyat Ruhiyat, Berlian Setiawaty, Muwafiqo Zamzami Dhuha

Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Jambura Journal of Mathematics has been indexed by

>>>More Indexing<<<

Creative Commons License

Jambura Journal of Mathematics (e-ISSN: 2656-1344) by Department of Mathematics Universitas Negeri Gorontalo is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. Powered by Public Knowledge Project OJS. 

Editorial Office

Department of Mathematics, Faculty of Mathematics and Natural Science, Universitas Negeri Gorontalo
Jl. Prof. Dr. Ing. B. J. Habibie, Moutong, Tilongkabila, Kabupaten Bone Bolango, Gorontalo, Indonesia