Multi-year non-life insurance risk of dependent lines of business in the multivariate additive loss reserving model

We derive analytical estimators of non-life insurance risk in multi-year view for the multivariate additive loss reserving model. Thereby we jointly assess reserve and premium risks of multiple years for portfolios of possibly dependent lines of business in one integrated approach. By extending existing formulae for the univariate additive model to the multivariate case, risk estimators for the aggregated portfolio now include the inherent dependencies among all lines of business. The resulting risk evaluation over one-year and general multi-year horizons is fundamental to regulatory reporting (e.g. the ORSA process in Solvency II) and risk-based business planning of non-life insurers with multiple lines of business. A case study illustrates the fruitful application of our formulae and reproduces previous findings for the special case of ultimo view.