Consider the problem of setting a provision for claims already incurred but not yet reported, or not fully paid. The past data used to construct estimates for the future payments consist of a triangle of incremental claims Xij.

The random variables Xij with i,j = 1,2,...,t denote the claim figures for year of origin i and development year j, meaning that the claims were paid in calendar year i+j-1. For (i,j) combinations with i+j=t+1, Xij has already been observed, otherwise it is a future observation. The purpose is to complete this run-off triangle to a square. For an introduction to claims reserving, we refer to the book Modern Actuarial Risk Theory (2001).

Using a log-Normal model, the first step is to transform the incremental claims by taking their logarithm. A model is then fitted to the transformed values using ordinary least squares regression analysis. An estimate for the Xij (i+j>t+1) is given by

where is the linear predictor and epsilon is a homoscedastic normally distributed random error component with zero mean. The loss reserve is defined as a high quantile of the random variable S, which is the discounted value of all future claim payments

where the discounting process is assumed to follow a Brownian motion

with µ and σ the yearly constant force of interest and the yearly volatility respectively.

Using the theory of comonotonic risks, one can calculate lower and upper bounds for the distribution of S. For more details, see Hoedemakers T., Beirlant J., Goovaerts M. and Dhaene J. (2002).