Univariate and Multivariate Claims Reserving with Generalised Link Ratios

In this paper, a regression modelling setting is introduced to estimate loss development factors, and its multivariate counterpart considers contemporaneous correlation between each regression equation within the triangle with homoscedastic or heteroscedastic errors, respectively. Using now an appropriate econometric framework, the prediction error is derived in a matrix form avoiding the calculation of the corresponding developments using computationally expensive recursive formulas. In this regard, the classical loss development factors method is extended to the univariate Generalized Link Ratios one, where the appropriate method selection is related with the minimization of the prediction errors in the triangle. In addition, the multivariate Generalized Link Ratios method is proposed with contemporaneous correlations between each regression equation within the triangle, using also the minimization of the prediction error as a way to select the appropriate method for the triangle. Mathematical expressions for the case of homoscedastic and heteroscedastic errors derive for some labelled methods (such as the chain ladder, vector projector and simple average) as well as for many other unnamed methods. Finally, several numerical examples with irregular, regular, and real data illustrate the applicability of our treatment and check the assumptions made in the paper.