Comments on Taylor's see-saw approach to claims reserving
Abstract Taylor (1981) introduces the See-Saw (SS) model for claims reserving in order to make allowance for speed of finalization. The model is applied to live data given in Taylor (1981) and there is prima facie evidence to suggest that it does well, especially in the light of comparisons of actual versus expected payments for each payment year. The purpose of the present paper is to demonstrate that from the point of view of operational forecasting, which is the object of the claims reserving exercise, the fitting of the linear SS can be improved upon. Moreover, we employ the SS as a vehicle for indicating the kind of validation tests that ought to be carried out once the parameters of a proposed model have been estimated from the data. Essentially, we indicate how the properties of the residuals may be used for diagnostic checking of the model. Many researchers involved in the claims reserving area are of the view that the data are extremely noisy especially if the model put forward only explains a small proportion of the total variation. Why not test whether this is the case? It turns out that the particular linear SS used by Taylor does not explain the signal accurately so that the balance of the variation is not entirely due to noise.
Speed of Finalization of Claims and Claims Runoff Analysis
To some extent, this paper provides theoretical back-up for TAYLOR'S (1980b) see-saw method of analysis of outstanding claims. Section 3 deals with the payments per claim finalized (PPCF) method. This assumes PPCF to be a function of development year only. Section 4 carries out a theoretical investigation of the validity of this assumption. It is found that the assumption is not justified in general; that PPCF can (and usually will) be sensitive to changes in speed of finalization of claims ; and that this sensitivity will depend on the convexity properties of PPCF as a function of development year. The analysis is made on the basis of the socalled hypothesis of invariant order. Section 5 develops the see-saw method from this theoretical basis, although Section 7 is at pains to point to situations in which that method remains applicable despite violation of the hypothesis of invariant order. Section 6 provides a numerical example.
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.