# Univariate and Multivariate Claims Reserving with Generalised Link Ratios

Insurance: Mathematics and Economics, Vol. 97, pp. 57-67, March 2021 DOI: 10.1016/j.insmatheco.2020.11.011

35 Pages Posted: 8 May 2019 Last revised: 28 Jan 2021

See all articles by Luis Portugal

ACTUARIAL Group

## Athanasios A. Pantelous

Monash University - Department of Econometrics & Business Statistics

## R. J. Verrall

City University London - The Business School

Date Written: November 27, 2019

### Abstract

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.

Keywords: Stochastic Reserving, Multivariate Regression, Homoscedastic and Heteroscedastic Errors, Seemingly Unrelated Regression, Prediction Errors

JEL Classification: G22, C13, C18, C35

Suggested Citation

Portugal, Luis and Pantelous, Athanasios A. and Verrall, Richard J, Univariate and Multivariate Claims Reserving with Generalised Link Ratios (November 27, 2019). Insurance: Mathematics and Economics, Vol. 97, pp. 57-67, March 2021 DOI: 10.1016/j.insmatheco.2020.11.011, Available at SSRN: https://ssrn.com/abstract=3370044 or http://dx.doi.org/10.2139/ssrn.3370044