统计研究 ›› 2018, Vol. 35 ›› Issue (1): 91-103.doi: 10.19343/j.cnki.11-1302/c.2018.01.010

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基于GB2分布的贝叶斯相依性准备金评估模型

李政宵 孟生旺   

  • 出版日期:2018-01-25 发布日期:2018-01-25

Bayesian Dependent Loss Reserving Models Based on GB2 Distribution

Li Zhengxiao & Meng Shengwang   

  • Online:2018-01-25 Published:2018-01-25

摘要: 非寿险精算的核心问题之一是对未决赔款准备金进行准确评估。非寿险未决赔款准备金评估通常使用增量赔款或累积赔款的流量三角形数据。在未决赔款准备金评估中,多条业务线的流量三角形数据之间通常存在一定的相依关系,这种相依关系对保险公司总准备金的评估结果具有重要影响。从本质上看,未决赔款准备金是一个随机变量,其损失分布存在一定的多样性。因此,在未决赔款准备金的评估中选择合适的分布至关重要。GB2分布是一种包含四个参数的连续型分布,具有灵活的密度函数,分布形状更加灵活,许多常见分布都是它的特例,适宜处理不同特点的未决赔款流量三角形数据。为了考虑不同业务线之间的相依关系对未决赔款准备金评估结果的影响,本文基于GB2分布建立了一种相依性准备金评估模型,该模型首先假设不同业务线的增量赔款服从GB2分布,并在分布的期望中引入事故年和进展年作为解释变量,引入日历年随机效应描述各条业务线之间的相依关系;然后借助贝叶斯HMC方法进行参数估计和未决赔款准备金预测,最后给出了总准备金的预测分布和评估结果。本文将该方法应用到两条业务线的流量三角形数据进行实证研究,并与现有其他方法进行了比较。实证研究结果表明,基于GB2分布的相依性准备金评估模型对未决赔款准备金的尾部风险和不确定性的考虑更加充分,更加适用于评估具有厚尾或者长尾特征的准备金数据。

关键词: 风险相依, GB2分布, 贝叶斯方法, 准备金评估

Abstract: One of the most critical problems in casualty insurance is to determine an appropriate outstanding reserve for incurred but unpaid losses. Forecasts and risk margins are often based on incremental or cumulative payment data corresponding to different business lines of loss triangles. Modeling dependency among multiple loss triangles has important implication for the determination of loss reserves in property and casualty insurance. In fact, owing to diversity of loss reserving data, it is critical to select the appropriate distribution. Generalized beta distribution of the second kind (GB2 distribution) has a flexible probability density function with four parameters, which nests various distributions with light and heavy tails, to facilitate accurate loss reserving in insurance applications. This paper proposes a Bayesian model based on GB2 distribution to capture dependence between two cells of two different runoff triangles. First, we use the GB2 distribution to fit the incremental paid losses data and introduce accident year and development year as covariates. Then, we suppose a dependence between all the observations that belong to the same calendar year (CY) for each line of business. This model can be done by using the calendar year as common random effect. For illustration, the model is applied to a dataset from Shi (2011) where a Bayesian method is proposed to estimate the distribution of the reserve. The result shows that the proposed model is more fully considered for the tail risk and uncertainty of the outstanding reserve than existing models, and is more suitable to model the loss reserving data with long and heavy tails.

Key words: Dependency, GB2 distribution, Bayesian Approach, Claims Reserve