Risk Measures of the ERNB Distribution Generated by G-NB Family

This paper provides VaR and CVaR risk measures, calculated for the Erlang-Negative Binomial (ERNB) distribution. The Erlang and negative binomial distributions are given and then ERNB distribution is obtained using a family of univariate distributions which is called G-Negative Binomial (G-NB). It is defined as compounding the negative binomial distribution (NB) with any continuous distribution (G). Here, we use the ERNB distribution obtained as taking Erlang distribution instead of G. In this paper, we focus on the estimation of VaR and CVaR risk measures for this distribution in closed form and the explicit expressions are also presented for some parameter values. The results are portrayed in the figures. In additionally, numerical examples are given to illustrate changing of the risk measure according to some parameters on a real data set of automobile insurance policies.

Keywords:

Erlang distiribution, the Erlang-Negative Binomial distribution, Value at Risk Conditional Value at Risk, G-NB family,

PDF

___

[1] Jorion, P. Risk management lessons from long term capital management. European financial management, 6 (2000) No.3, 277-300.
[2] Denuit M., Dhaene J. and, Goovaerts M.J., Kaas R. Actuarial Theory for Dependent Risks; Measures, Orders and Models, John Wiley and Sons, 2005.
[3] Rockafellar, R. T., and Uryasev, S. . Conditional value-at-risk for general loss distributions. Journal of banking & finance, 26(2002), 7, 1443-1471.
[4] Artzner, P., Delbaen, F., Eber, J. M., and Heath, D. . Coherent measures of risk. Mathematical finance, 9 (1999),3, 203-228.
[5] Embrechts, P., Resnick, S. I., and Samorodnitsky, G. Extreme value theory as a risk management tool. North American Actuarial Journal, 3(1999), 2, 30-41.
[6] Pflug, G. Some Remarks on the Value at Risk and the Conditional Value at Risk, in: S.P. Uryasev (ed.), Probabilistic Constrained Optimization: Methodology and Applications. Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 272-281, 2000.
[7] Krokhmal, P., Palmquist, J., and Uryasev, S. Portfolio optimization with conditional value-at-risk objective and constraints. Journal of risk, 4 (2002), 43-68.
[8] Jiménez, J. A., and Arunachalam, V. . Using Tukey’sg and h family of distributions to calculate value-at-risk and conditional value-at-risk. Journal of Risk, 13 (2011), 4, 95-116.
[9] Embrechts, P., Kluppelberg, S., and Mikosch, T. Extremal events in finance and insurance, 1997.
[10] Rau-Bredow, H. Value at risk, expected shortfall, and marginal risk contribution. Risk Measures for the 21st Century, Szego, G.(ed.), Wiley Finance, 61-68, 2004.
[11] Cummins, J. D., Dionne, G., McDonald, J. B., and Pritchett, B. M. Applications of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics and Economics, 9 (1990), 4, 257-272.
[12] Dickson, D. C., and Hipp, C. On the time to ruin for Erlang (2) riskprocesses. Insurance: Mathematics and Economics, 29(2001), 3, 333-344.
[13] Yuen, K. C., Guo, J., and Wu, X. On a correlated aggregate claims model with Poisson and Erlang risk processes. Insurance: Mathematics and Economics, 31(2002), 2, 205-214.
[14] Lefèvre, C., and Picard, P. Appell pseudopolynomials and Erlang-type risk models. Stochastics An International Journal of Probability and Stochastic Processes, 86 (2014), 4, 676-695.
[15] Straub, E. Swiss Association of Actuaries (Zürich). Non-life insurance mathematics (No. 517/S91n). Berlin: Springer, 1988.
[16] Chen, S. X. Probability density function estimation using gamma kernels. Annals of the Institute of Statistical Mathematics, 52 (2000), 3, 471-480.
[17] Lee, S. C., and Lin, X. S. Modeling and evaluating insurance losses via mixtures of Erlang distributions. North American Actuarial Journal, 14 (2010), 1, 107-130.
[18] Jeon, Y., and Kim, J. H. A gamma kernel density estimation for insurance loss data. Insurance: Mathematics and Economics, 53 (2013), 3, 569-579.
[19] Pascal B. Varia opera mathematica D.Petri de Fermat, Tolossae,1679.
[20] Student. On the error of counting with a haemacytometer. Biometrika, (1907), 351-360.
[21] Gómez-Déniz, E., Sarabia, J. M., and Calderín-Ojeda, E. Univariate and multivariate versions of the negative binomial-inverse Gaussian distributions with applications. Insurance: Mathematics and Economics, 42(2008), 1, 39-49.
[22] Pudprommarat, C., Bodhisuwan, W., and Zeephongsekul, P. . A new mixed negative binomial distribution. Journal of Applied Sciences, 12 (2012),17, 1853.
[23] Ramos, M. W. A., Percontini, A., Cordeiro, G. M., and da Silva, R. V. The Burr XII negative binomial distribution with applications to lifetime data. International Journal of Statistics and Probability, 4 (2015), 1, 109.
[24] Kongrod S., Bodhisuwan W., Payakkapong P. The negative binomial-Erlang distribution with applications Introduction Journal of Pure and Applied Mathematics, 92 (2014), 3, 389-401.
[25] Percontini, A., Cordeiro, G. M., and Bourguignon, M. The G-Negative Binomial Family: General Properties and Applications. Advances and Applications in Statistics, 35 (2013), 127-160.
[26] Erlang A.K. Solution of some problems in the theory of probabilities of significance in automatic telephone exchanges. Elektrotkeknikeren, 13 (1917), 513.

Mathematical Sciences and Applications E-Notes-Cover

ISSN: 2147-6268
Yayın Aralığı: Yılda 4 Sayı
Başlangıç: 2013
Yayıncı: -

Arşiv