The sensitivity of the ruin probability depending on the claim size distribution has been the topic of several discussion papers in recent ASTIN Bulletins. This discussion was initiated by a question raised by Schmitter at the ASTIN Colloquium 1990 and attempts to make further contributions to this problem. We find the necessary and sufficient conditions for fitting three given moments by diatomic and diexponential distributions. We consider three examples drawn from fire (large spread), individual life (medium spread) and group life (small spread) insurance data, fit them with diatomics and diexponentials whenever the necessary and sufficient conditions are met, and compute the ruin probabilities using well known formulas for discrete and for combination of exponentials claim amounts. We then compare our approximations with the exact values that appeared in the literature. Finally we propose using diatomic and diexponential claim distributions as tools to study the Schmitter problem.

Stimulated by a recent contribution by G. Venter in this journal the adequate-ness of (re-)insurance premium calculation based on the hypothesis of arbitrage free (re-)insurance markets is questioned. It is argued that—in contrast to the theory of financial markets—it is not reasonable to demand that insurance markets are arbitrage free. In addition the adjusted distribution principles put forward by Venter are claimed to be invalid.

Albrecht's interesting discussion raises two major points: is an arbitrage free insurance market a reasonable assumption, and are adjusted distribution principles appropriate?

I had assumed an arbitrage free insurance market without really discussing reasons in support of it. Certainly there have been arbitrage profits made through reinsurance, possibly, for example, by selling excess on excess at rates based on regular excess policies, and perhaps in the London Market Excess arena. But even if arbitrage occurs frequently, I feel that an arbitrage free market is still a reasonable assumption for theoretical developments, if we take it to mean that systematic ongoing arbitrage is not possible. Although some arbitrage situations have probably lasted a few years, all I am aware of eventually have disappeared as they became better known. Either entrepreneurs try to outbid each other to take advantage of them, or the sellers lose their willingness to play. A list of insurers that are regularly making arbitrage profits would be an interesting refutation of this assumption.

Albrecht gives some examples of the well known and quite true proposition that larger portfolios are more stable. Large insurers can maintain the same security level at a lower price. Our risk theory training leads actuaries to believe that the smaller needed security premium for large insurers will induce them to charge lower prices. This is not necessarily true in the market, however. Larger insurers may in fact charge the market price and make more profits, for example. Also, it would be fairly surprising if small insurers actually do get away with charging more for the same risks. If they do, they should buy prorata reinsurance heavily from the large insurers at the large insurers' risk price and pocket the difference. There are of course transaction costs to buying prorata reinsurance; those may in fact be the limit of what the smaller insurers can demand as the supposed bonus price they get from the market for being small.