Premium Calculation Without Arbitrage Author's Reply on the Note by P. Albrecht

Abstract

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.