We introduce a stochastic model for the development of attritional and large claims in long-tail lines of business and present a corresponding "chain ladderlike" IBNR method which allows the use of claims payment data for attritional and claims incurred data for large losses. We derive formulas for the mean squared error of prediction and apply the method to a German motor third party liability portfolio.KEYWORDS IBNR method, chain ladder, large loss, attritional loss, mean squared error of prediction, standard error.

Pricing approaches for long-tail quota shares are often based on the chain ladder method. Apart from IBNR calculation, common pricing methods require volume measures for accident years in the observation period, and for the quotation period. In practice, in most cases restated premiums are used as the volume measures. The prediction error of the chain ladder method is an important part of the prediction uncertainty of these pricing approaches. There are, however, two sources of uncertainty that are not addressed by the chain ladder model: the stochastic volatility of the claims in the first development year; and the restatement uncertainty, the risk that the restated premium is not a good volume measure. We extend Mack's chain ladder model to cover these two sources of uncertainty, and calculate the mean-squared error of chain ladder pricing approaches with arbitrary weights for the accident years in the observation period. Then we focus on the problem of finding optimal weights for the accident years. First, we assume that the parameters for restatement uncertainty are given, and provide recursion formulas to calculate approximately-optimal weights. Second, we describe a maximum likelihood approach that can be used to estimate the restatement uncertainty.

No abstract

To a topological space V we assign the bordism group N def n (V ) of regularly defective maps f : M •→V on closed n-dimensional manifolds M . These are triples (M, ∆, f ) where ∆ is a closed submanifold ∆ ⊂ M and f a continuous map f : M ∆ → V . We briefly review the construction of the defect complex DV given by M. Rost in [17] and show that N def n (V ) is isomorphic to ordinary bordism N n (DV ). The bordism classes in N def n (V ) ∼ = N n (DV ) are detected by characteristic numbers twisted with cohomology classes of DV . Some of these numbers can be described without reference to the defect complex. As an example we treat the case of the circle V = S 1 . We compute N def n (S 1 ), construct a basis and a complete set of characteristic numbers.

No abstract