Modelling Dependence between Loss Triangles using Copula and DVine Constructions
Matthias Lindinger · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2010
Gaussian Copula Model NotationThis section will provide the reader with the basic notation used throughout the following article.• Loss triangle: the m different loss triangles are indexed by l = 1, . . ., m , where each l stands for only one triangle.• Observation period: the length of the time period in which the data has been observed is denoted by n.• Accident year i: the year in which an accident, the company has to pay for, occurs is indexed with i , i = 1, 2, . . ., n.Thus if i = 1 the accident year is the year when the data collection started, . . ., if i = n the accident year is the last year for which data is available.• Development year j: Payments made for accidents in the year they occur are, by definition, settled in development year j = 0 , payments made one year after the accident in development year j = 1,. . .,payments made in the kth year following the accident in development year j = k, and so on.Therefore the index j representing the development year runs from j = 0, . . ., n -1.• Calendar year t: the calendar years are denoted by t = 1, ..., n, where t = 1 corresponds to the calendar year in which data collection started , . . ., t = n to the last calendar year of data collection.• The triangle entries p lij are the sum of all payments made for accidents occured in accident year i, i = 1, . . ., n in the jth year after those accidents in loss triangle l, l= 1, . . ., m.This notation might seem difficult for readers who are not familiar with loss triangles.Therefore, I will use a simple example : Consider a time period of n = 5 years, i.e, the period from 2005 to 2009, where payments for a certain business line (triangle) l have been obvserved.following from α lt ∼ N (0, 1) and ǫ lij ∼ N (0, 1), l = 1, . . ., m.•