Estimation of outstanding reinsurance recoveries on the basis of incomplete information

The paper considers the following situation. The actuary is asked to estimate the outstanding claims of an insurance portfolio, both gross and net of (excess-of-loss) reinsurance. He obtains a gross estimate in the usual way by reference to historical information in respect of gross payments. He must then transform this to a net estimate. The only information available to him as regards the relation between gross and net outstanding comprises: 1. (i) the number of claims outstanding; 2. (ii) total of physical estimates of gross amounts outstanding; 3. (iii) total of physical estimates of net amounts outstanding. The paper considers various net estimates which can be derived from this very restricted information. Basically, the approach is that if the actuary's gross estimate is higher (lower) than the corresponding physical, then so should be his net estimate. In Section 3, it is shown that it is wrong to adjust from gross to net simply in proportion with the physical estimates. Section 4 considers how, for a given physical estimate of the net-lo-gross ratio, the actuary's estimate of this ratio should respond to any information concerning the danger or the failure rate of the distribution of individual claim size. Section 5 considers a couple of simple devices for numerical evaluation of the net-to-gross ratio. The simplest device involves the assumption of a negative exponential distribution of individual claim sizes. This produces a particularly simple result which can be expressed in a form not involving the excess-of-loss retention level. A 2-parameter family of distributions is then introduced encompassing the negative exponential (constant failure rate) and in addition increasing and decreasing failure rates. The last case is most relevant and is analysed. A reasonably simple computation produces the net-to-gross ratio. Section 6 considers extremal claim size distributions, i.e. those which produce the maximal reinsurance recoveries consistent with the physical estimate of net-to-gross ratio. Again simple results are obtained. The paper concludes, in Section 7, with a numerical example drawn from actual experience. Author Keywords: Outstanding claims; Excess-of-loss reinsurance; Gross estimates; Net estimates; Reinsurance recoveries; Retention limit; Failure rate *1 This paper was presented to a one-day seminar on Insurance Mathematics at the H.C. Oersted Institute, University of Copenhagen, Denmark, June 1981.