손해보험사의 재보험료 산출을 위한 손해분포의 적합

박상균 ( Sangkyun Park ),송성주 ( Seongjoo Song ) 한국보험학회 2019 保險學會誌 Vol.119 No.-

본 논문에서는 손해보험사의 다양한 보험종목 손해액의 개별적인 특성을 고려하면서 동시에 그들 사이에 존재하는 의존성을 반영하여 손해액의 합에 대한 결합 재보험료를 산출하고자 하였다. 의존성 반영을 위해 코퓰라 함수를 이용하였고, 각 보험종목 손해액에 대한 주변분포로는 normal inverse Gaussian 분포와 일반화 파레토분포, 와이블 분포를 사용하였다. 모의실험과 삼성화재, 현대해상의 월별 손해액 자료를 통해 단일 보험종목 재보험료 산출결과를 살펴보고, 삼성화재와 현대해상 자료에 대한 다변량 분포를 적합하여 재보험료를 산출하였다. 보험종목 간의 의존성을 고려한 경우 재보험료가 크게 추정되었으며, 두꺼운 꼬리를 잘 설명할 수 있는 일반화 파레토분포와 normal inverse Gaussian 분포를 사용하였을 때 정규분포나 와이불 분포를 사용했을 때에 비해 높은 재보험료가 얻어졌다. Normal inverse Gaussian 분포는 일변량 손해액 자료에 높은 적합도를 보이면서 일반화 파레토분포와 크게 다르지 않는 재보험료를 산출하고 있으며, 일반화 파레토분포와 달리 임계치를 따로 설정할 필요가 없고 수치적분도 상대적으로 빠르게 수행되어 보다 편리하게 사용할 수 있었다. In this paper, we calculate the reinsurance premiums on the sum of several insurance lines for non-life insurance companies considering the dependency among loss amounts of lines as well as their individual characteristics. We try several copula functions to reflect the dependency among loss amounts, and apply normal inverse Gaussian distribution, generalized Pareto distribution, and Weibull distribution for the marginal distribution of the monthly loss amount of each insurance field. Through simulation studies and real data analysis with loss amounts from Samsung Fire&Marine Insurance and Hyundai Insurace, we calculate the reinsurance premiums for individual insurance field. Also, we calculate enterprise-wide reinsurance premiums by fitting multivariate distributions to monthly losses from Samsung Fire&Marine Insurance and Hyundai Insurace. When we take into account the dependence structure among different insurance fields, reinsurance premiums were calculated greater compared to those based on the independence assumption. Moreover, the generalized Pareto distribution and the normal inverse Gaussian distribution provided larger premiums than the normal or Weibull distribution when they were used as marginal loss distribution . The normal inverse Gaussian distribution yields a good fit to the marginal loss data and produces premiums that do not differ significantly from the generalized Pareto distribution. Unlike the generalized Pareto distribution, there is no need to set a threshold value, which makes it more convenient to use.

RECURRENCE RELATIONS FOR QUOTIENT MOMENTS OF GENERALIZED PARETO DISTRIBUTION BASED ON GENERALIZED ORDER STATISTICS AND CHARACTERIZATION

Kumar, Devendra Chungcheong Mathematical Society 2014 충청수학회지 Vol.27 No.3

Generalized Pareto distribution play an important role in reliability, extreme value theory, and other branches of applied probability and statistics. This family of distribution includes exponential distribution, Pareto or Lomax distribution. In this paper, we established exact expressions and recurrence relations satised by the quotient moments of generalized order statistics for a generalized Pareto distribution. Further the results for quotient moments of order statistics and records are deduced from the relations obtained and a theorem for characterizing this distribution is presented.

A Modified Hybrid Gamma and Generalized Pareto Distribution for Precipitation Data

김용구,Hyeongang Kim,이규원,민기홍 한국기상학회 2019 Asia-Pacific Journal of Atmospheric Sciences Vol.55 No.4

This study introduces a modified hybrid gamma and generalized Pareto distribution. Prior to this, we define a general spliced distribution and its corresponding gamma distribution, which is part of the head, and a generalized Pareto (GP) distribution, which is part of the tail. We then examine the threshold conditions for the modified hybrid gamma and GP distribution and defined probability density function. Also, we derive the negative log-likelihood function of the modified hybrid gamma and GP distribution and estimate approximate maximum likelihood estimates using the differential evolution algorithm for each simulation to minimize it. Moreover, by presenting the mean square error for each sample size, the model is evaluated according to the size of the sample. Finally, we use daily observed summer precipitation for Seoul, Korea, from 1961 to 2011, which includes 4692 data sets.We use 2051 data sets corresponding to wet conditions. As a result, the estimated threshold of the modified hybrid gamma and GP distribution is 0.1455. After deriving Fisher information through the Hessian matrix, we also present the standard error of the maximum likelihood estimator.

논문 : 금융시장에서 발생하는 극단적인 사건의 모형화와 손실함수의 추정

이호진 ( Hojin Lee ) 명지대학교 금융지식연구소 2016 금융지식연구 Vol.14 No.1

본 연구에서는 1871년 1월부터 2013년 1월까지의 S&P 500 지수수익률로부터 발생하는 극단적인 손실값의 분포를 추정하고, 이를 이용하여 극단적인 손실이 발생할 확률과 일정 수준이상의 손실이 발생하는 경우의 수익률을 계산할 수 있음을 보이고 있다. 연간 수익률 자료를 이용하여 표본시계열 이후의 일정 기간 동안 극단적인 손실이 발생할 확률을 계산하거나, 일정 수준 이상의 손실이 발생하는 경우의 수익률을 계산하기 위하여 일반적인 극단값 본포(Generalized Extreme Value Distribution)를 이용하였다. 또한 일반화된 파레토 분포(Generalized Pareto Distribution)을 추정하여, VaR (Value-at-Risk)이나 기대손실 (Expected Shortfall)을 계산하는 방법을 이용한 후, 두 가지 계산방법에 의해 계산된 위험척도를 비교분석하였다. 두 가지 방법 중 어떠한 방법을 사용하고, 어떠한 위험척도를 이용하더라도, 지수 수익률의 정규분포를 가정하는 방법론은 포트폴리오의 위험수준을 과소평가하는 것으로 나타났다. We estimate the extreme value distribution of the losses on the monthly S&P 500 index returns during the period from January 1871 to January 2013 to quantify the tail probability and the extreme quantile of the loss distribution. In estimating the generalized extreme value distribution (GEV), we use the Fisher-Tippett theorem in specifying the limiting distribution for centered and normalized maxima. We use the estimated extreme value distribution to calculate the probability of observing an unprecedented annual maximum loss on the stock market index over the next period. We also compute the return level which is exceeded by the annual maximum negative return in any particular year with a given level of probability. As an alternative to the GEV distribution estimation with the block maxima data extracted from the sample of the disjoint t ime period, we estimate t he l imiting distribution o f scaled e xcesses ov er a h igh threshold to improve the efficiency of the GEV estimation. We estimate the generalized Pareto distribution (GPD) and compute the Value-at-Risk (VaR) and the expected shortfall (ES) and confirm that the risk measures under the normal distribution underestimate the extreme quantile estimation.

ON RELATIONS FOR QUOTIENT MOMENTS OF THE GENERALIZED PARETO DISTRIBUTION BASED ON RECORD VALUES AND A CHARACTERIZATION

Kumar, Devendra The Korean Society for Computational and Applied M 2013 Journal of applied mathematics & informatics Vol.31 No.3

Generalized Pareto distributions play an important role in re-liability, extreme value theory, and other branches of applied probability and statistics. This family of distribution includes exponential distribution, Pareto distribution, and Power distribution. In this paper we establish some recurrences relations satisfied by the quotient moments of the upper record values from the generalized Pareto distribution. Further a char-acterization of this distribution based on recurrence relations of quotient moments of record values is presented.

ON RELATIONS FOR QUOTIENT MOMENTS OF THE GENERALIZED PARETO DISTRIBUTION BASED ON RECORD VALUES AND A CHARACTERIZATION

Devendra Kumar 한국전산응용수학회 2013 Journal of applied mathematics & informatics Vol.31 No.3

Generalized Pareto distributions play an important role in reliability, extreme value theory, and other branches of applied probability and statistics. This family of distribution includes exponential distribution, Pareto distribution, and Power distribution. In this paper we establish some recurrences relations satisfied by the quotient moments of the upper record values from the generalized Pareto distribution. Further a characterization of this distribution based on recurrence relations of quotient moments of record values is presented.

Acceptance Sampling Plan for Truncated Life Tests Based on Generalized Pareto Distribution using Mean Life

Navjeet Singh,Anju Sood,Navyodh Singh,G.S. Buttar 대한산업공학회 2020 Industrial Engineeering & Management Systems Vol.19 No.3

Under truncated life tests, two types of attribute acceptance sampling plans are proposed, first is sequential sampling plan and other is repetitive acceptance sampling plan. These plans ensure the quality of products in terms of the mean lifetime when the lifetime follows the generalized Pareto distribution. Time duration of completing the experiment plays a vital role to finalize the acceptance sampling plan. The sample size is directly connected with the time duration of completing the experiment, so whichever plan will give the most suitable unknown variable (sample size) or average sample number under the specified values of producer’s risk and consumer’s risk will be regarded as more convenient. The average sample numbers of sequential sampling plan are calculated for different values of shape parameter, consumer’s risk and producer’s risk. We extant a simulation study to help the proposed techniques and a comparison between the repetitive acceptance sampling plan and sequential sampling plan is made. Furthermore, we present a comparative study of proposed plan with Rasay et al. (2018). For the proposed sampling plan some useful tables have been developed for practical utilization.

Extreme value modeling of structural load effects with non-identical distribution using clustering

Junyong Zhou,Xin Ruan,Xuefei Shi,Chudong Pan 국제구조공학회 2020 Structural Engineering and Mechanics, An Int'l Jou Vol.74 No.1

The common practice to predict the characteristic structural load effects (LEs) in long reference periods is to employ the extreme value theory (EVT) for building limit distributions. However, most applications ignore that LEs are driven by multiple loading events and thus do not have the identical distribution, a prerequisite for EVT. In this study, we propose the composite extreme value modeling approach using clustering to (a) cluster initial blended samples into finite identical distributed subsamples using the finite mixture model, expectation-maximization algorithm, and the Akaike information criterion; (b) combine limit distributions of subsamples into a composite prediction equation using the generalized Pareto distribution based on a joint threshold. The proposed approach was validated both through numerical examples with known solutions and engineering applications of bridge traffic LEs on a long-span bridge. The results indicate that a joint threshold largely benefits the composite extreme value modeling, many appropriate tail approaching models can be used, and the equation form is simply the sum of the weighted models. In numerical examples, the proposed approach using clustering generated accurate extrema prediction of any reference period compared with the known solutions, whereas the common practice of employing EVT without clustering on the mixture data showed large deviations. Real-world bridge traffic LEs are driven by multi-events and present multipeak distributions, and the proposed approach is more capable of capturing the tendency of tailed LEs than the conventional approach. The proposed approach is expected to have wide applications to general problems such as samples that are driven by multiple events and that do not have the identical distribution.

Some Exponentiated Distributions

Ali, M. Masoom,Pal, Manisha,Woo, Jung-Soo The Korean Statistical Society 2007 Communications for statistical applications and me Vol.14 No.1

In this paper we study a number of new exponentiated distributions. The survival function, failure rate and moments of the distributions have been derived using certain special functions. The behavior of the failure rate has also been studied.

Modeling Non-Normally Distributed Stock Portfolio Returns and Applications to Risk Management

이호진 한국계량경제학회 2015 계량경제학보 Vol.26 No.3

We utilize the copula function methodology to separate out the components which describe the marginal behavior of the return processes and the dependence structure between the random variables from the joint density. In order to reflect the non-ellipticity of the joint distribution and heavy tails in the extreme quantile of the marginal distributions of asset returns, we use the generalized Pareto distribution (GPD) as the margins and a variety of parametric copula functions along with a nonparametric copula function in the analysis. We select the optimal copulas from a variety of non-nested copulas based on the model selection criteria. In calculating the risk measures, we assume that the returns are jointly distributed to the parametric copulas as well as to the empirical copula. We then compare the result with that from the bivariate normal distribution. The results show that the VaR and ES computed from the copula function which takes the complicated and possibly nonlinear dependence structure into account performs better than the one based on the linear correlation-based normality assumption.