Modeling Dependence between Loss Triangles

A critical problem in property and casualty insurance is forecasting incurred but as yet unpaid losses. Forecasts and risk margins are often based on individual loss triangles with each triangle corresponding to a different line of business. Different lines of business are often dependent, and overall risk margins must reflect this dependence. This article develops, implements, and applies a model for loss triangle dependence. The model facilitates the structuring and measurement of dependence. One possible structure is where payments in different triangles in the same calendar year are related. Dependence is modeled with a Gaussian copula, and it is moderated by quantities called communalities that measure the relative impact of cross-dependence in each triangle. Dependence can be structured in terms of factor models. Methods reduce to relatively simple calculations in the case of marginal normal distributions. Procedures are applied to U.S. loss triangle data. The impact of loss triangle dependence on risk margins is considered.