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Density approximations and VaR computation for compound Poisson-lognormal distributions. (English) Zbl 1364.62038
Summary: Parametric approximations of the compound Poisson-lognormal distribution are developed and used to compute Value-at-Risk (VaR). As guidelines for finding an approximation, the skewness-kurtosis space and the tail behavior are considered. The Generalized Beta distribution of the second kind (GB2) and a mixture of lognormals are found to provide a good fit. In certain cases, the GB2 can be estimated by moment-matching, thus providing a simulation-free procedure for VaR computation. For confidence levels larger than 99%, extreme value theory approaches are developed. According to extensive Monte Carlo evidence, the proposed approximations are more efficient than crude Monte Carlo.

MSC:
62E17 Approximations to statistical distributions (nonasymptotic)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Software:
evir; nleqslv; QRM
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References:
[1] DOI: 10.1287/ijoc.7.1.36 · Zbl 0821.65085
[2] Bee M., The Advanced Measurement Approach to Operational Risk pp 123–
[3] Boland P., Statistical and Probabilistic Methods in Actuarial Science · Zbl 1124.62069
[4] Embrechts P., Modelling Extremal Events for Insurance and Finance · Zbl 0798.90024
[5] Faádi Bruno F., Annali di Scienze Matematiche e Fisiche 6 pp 479–
[6] Faádi Bruno F., The Quarterly Journal of Pure and Applied Mathematics 1 pp 359–
[7] DOI: 10.1080/07350015.2013.847842
[8] Glasserman P., Monte Carlo methods in financial engineering · Zbl 1038.91045
[9] Graf M., GB2: Generalized Beta Distribution of the Second Kind: properties, likelihood, estimation
[10] DOI: 10.1016/j.ejor.2004.06.012 · Zbl 1079.90009
[11] Hasselman B., Nleqslv: Solve systems of non linear equations · Zbl 1346.65034
[12] DOI: 10.1080/00401706.1987.10488243
[13] DOI: 10.1103/PhysRev.106.620 · Zbl 0084.43701
[14] Johnson N. L., Univariate Discrete Distributions, 3. ed.
[15] Johnson N. L., Univariate Continuous Distributions, 2. ed.
[16] Kapur J., Maximum Entropy Models in Science and Engineering · Zbl 0746.00014
[17] Kleiber C., Statistical Size Distributions in Economics and Actuarial Sciences · Zbl 1044.62014
[18] Klugman S. A., Loss Models: From Data to Decisions, 2. ed. · Zbl 1159.62070
[19] DOI: 10.1029/WR015i005p01055
[20] DOI: 10.2307/1913469 · Zbl 0557.62098
[21] DOI: 10.1111/j.1475-4991.2011.00478.x
[22] McNeil A., Quantitative Risk Management: Concepts, Techniques, Tools · Zbl 1337.91003
[23] Moscadelli M., The Modelling of Operational Risk: Experience With the Analysis of the Data Collected by the Basel Committee, Vol. 517 of Temi di discussione
[24] DOI: 10.1017/S0269964813000053 · Zbl 1275.60021
[25] Panjer H. H., Operational Risk Modeling Analytics · Zbl 1258.62101
[26] Pfaff B., Evir: Extreme Values in R
[27] Wolny-Dominiak A., insuranceData
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