In the wake of the technical development of our era we are increasingly faced with claims of extremely large amounts.

Whereas even in the past century large claims were mostly due to elemental or natural forces such as earthquakes, hurricanes and floods, there are today other causes, conditioned by human factors, that have considerably increased in importance. Another fact to be observed is that the risks that are passed on to the insurance market assume a more serious character year by year. Large building complexes, giant tankers, containers, atomic power stations, dams and jumbo jets pose for the insurers problems which are in no way easy to solve, since the risk covers required touch the limits of market capacity and their rating is subject to great uncertainties. At the same time, the readiness to underwrite such covers is a matter of utmost importance for the private insurance industry.

For the coverage of such large risks, various insurance forms have been developed in the past years within the range of non-proportional methods. Besides the excess of loss and stop loss covers in their usual form, the cumulative loss cover and the largest claim cover have attracted particular attention. The cumulative risk cover has, for example, been dealt with in [11]) and has also been frequently offered in practice. The coverage of the largest claim or, more generally, of the sum of the n largest claims does not appear to have gained a proper foothold in practice in spite of various valuable contributions in this field—considering particularly [2], [3], [5], [8] and [14]. The fact that it was during 1963/64 that various authors analysed these problems is no coincidence, for it was then that in two ASTIN Colloquia these topics were discussed.