Value at Risk Estimation using the Kappa Distribution with Application to Insurance Data

Document Type : Original Article


Department of Mathematics and Statistics, Lahijan branch, Islamic Azad University, Lahijan, Iran


The heavy tailed distributions have mostly been used for modeling the financial data. The kappa distribution has higher peak and heavier tail than the normal distribution. In this paper, we consider the estimation of the three unknown parameters of a Kappa distribution for evaluating the value at risk measure. The value at risk (VaR) as a quantile of a distribution is one of the important criteria for financial institution risk management. The maximum likelihood, moment, percentiles and maximum product of spacing methods are considered to estimate the unknown parameters. The data of the insurance stock prices is analyzed for comparing the proposed methods in VaR evaluation. An important implication of the present study is that the Kappa distribution can be considered as a loss distribution for the VaR estimation. Also, it is observed that the maximum likelihood estimator, in contrast to other estimators, provides smallest VaR in the proposed stock prices data.


1)       Abada, P.,  Benitob, S., Lópezc, C. (2014). A comprehensive review of Value at Risk methodologies. The Spanish Review of Financial Economics, 12, 1-46.
2)         Ashour, S.K., Elsherpieny, E.A., Abdelall, Y.Y. (2009). Parameter Estimation for Three-Parameter Kappa Distribution under Type II Censored Samples. Journal of Applied Sciences Research, 5(10), 1762-1766.
3)       Basu, B., Tiwari, D., Kundu, D., Prasad, R. (2009). Is Weibull distribution the most appropriate statistical strength distribution for brittle materials?. Ceramics International, 35, 237-246.
4)       Brandolin, D., Colucci, S. (2012). Backtesting value-at-risk: a comparison between filtered bootstrap and historical simulation. Journal of Risk Model Validation, 13, 3-16.
5)       Braione M., Scholtes, N.K. (2016). Forecasting Value-at-Risk under different distributional assumptions. Econometrics, 4, 1-27.
6)         Cheng, R. C. H., Amin, N. A. K. (1983). Estimating parameters in continuous univariate distributions with a shifted origin. Journal of the Royal Statistical Society Series B (Methodological), 45, 394-403.
7)       Čorkalo, S. (2011). comparison of value at risk approaches on a stock portfolio. Croatian Operational Research Review (CRORR), 2, 81-90.
8)         Dupuis, D.J., Winchester, C. (2007). More on the four-parameter Kappa distribution. Journal of Statistical Computation and Simulation, 71, 99-113.
9)         Dupuis, D.J., Papageorgiou, N., Rémillard, B. (2015). Robust Conditional Variance and Value-at-Risk Estimation. Journal of Financial Econometrics, 13:896–921.
11)     Gebizlioglu, O.L., Şenoğlu, B., MertKantar, Y. (2011). Comparison of certain value-at-risk estimation methods for the two-parameter Weibull loss distribution. Journal of Computational and Applied Mathematics, 11, 3304-3314.
12)    Hang, A. (2009). Value at risk estimation by quantile regression and kernel estimator.  Review of Quantitative Finance and Accounting, 19(5),379-395.
13)      Jeng, B.Y., Murshed, Md.S., Seo, Y.A., Park, J.S. (2014). A three-parameter Kappa distribution with hydrologic application: a generalized Gumbel distribution. Stochastic Environmental Research and Risk Assessment, 28, 2063–2074.
14)    Johnson, B.A., Long, Q., Huang, Y., Chansky, K., Redman, M. (2016). Model selection and inference for censored lifetime medical expenditures. Biometrics, 72(3),731-41.
15)      Kao, J.H.K. (1958). Computer methods for estimating Weibull parameters in reliability studies. IRE Transactions on Reliability and Quality Control, 13,15–22.
16)    Kim, J. (2015). Heavy Tails in Foreign Exchange Markets: Evidence from Asian Countries. Journal of Finance and Economics, 3, 1-14.
17)      Kjeldsen, T.R., Ahn, H., Prosdocimi, L. (2017). On the use of a four-parameter Kappa distribution in regional frequency analysis. Hydrological Sciences Journal, 62, 1354-1363.
18)      Kumphon, B. (2012). Maximum Entropy and Maximum Likelihood Estimation for the Three-Parameter Kappa Distribution. Open Journal of Statistics, 2,415-419.
19)    Livadiotis, G., McComas, D.J. (2013). Understanding Kappa Distributions: A Toolbox for Space Science and Astrophysics. Space Science Reviews, 175, 183–214.
20)     Mentel, G. (2013). Parametric or Non-Parametric Estimation of Value-At-Risk. International Journal of Business and Management, 8,103-112.
21)      Nwobi, F.N., Ugomma, C.A. (2014). A Comparison of Methods for the Estimation of Weibull Distribution Parameters. Metodološki zvezki, 11, 65-78.
22)      Ouarda, T.B.M.J., Charron, C., Shin,J.Y., Marpu, P.R., Al-Mandoos, A.H., Al-Tamimi, M.H., Ghedira, H., Al Hosary, T.N. (2015). Probability distributions of wind speed in the UAE. Energy Conversion and Management, 93, 414-434.
23)      Panahi, H. (2016). Model Selection Test for the Heavy-Tailed Distributions under Censored Samples with Application in Financial Data. International Journal of Financial Studies, 4,1-14.
24)     Panahi, H. (2017). Estimation Methods for the Generalized Inverted Exponential Distribution under Type II Progressively Hybrid Censoring with Application to Spreading of Micro-Drops Data.  Communications in Mathematics and Statistics, 5,159-174.
25)    Pierrard, V.,  Lazar, M. (2010). Kappa Distributions: Theory and Applications in Space Plasmas. Solar Physics, 267, 153-174.
26)     Singh, S., Maddala, G. (1976). A Function for the Size Distribution of Income. Econometrica, 44, 963-970.
27)    Sinha, P., Agnihotri, S. (2015). Impact of non-normal return and market capitalization on estimation of VaR.  Journal of Indian Business Research, 7, 222-242.
28)      Swami, O.S., Pandey, S.K., Pancholy, P. (2016). Value-at-Risk Estimation of Foreign Exchange Rate Risk in India. Asia-Pacific Journal of Management Research and Innovation, 12(1),1–10.
29)    Wong, Z.Y., Chin, W.C., Tan, S.H. (2016). Daily value-at-risk modeling and forecast evaluation: The realized volatility approach. The Journal of Finance and Data Science, 2, 171-187