Remarks on the Swiss premium principle on positive risks
Abstract In this note it is proved that the Swiss premium principle is completely characterized by its premiums for positive Bernoulli risks. This yields another interpretation of the Swiss premium principle.
On Quasi-mean value principles
ZusammenfassungEin neues Prämienprinzip, das Fast-Mittelwertprinzip, mit dem endogenen Fast-Mittelwertprinzip als Spezialfall, wird eingeführt. Es beschreibt eine Anzahl von bekannten Prämienprinzipien wie das Schweizer Prinzip, das Prinzip von Niederlande, das Perzentilprinzip, das Prinzip von Karlsruhe, das Mittelwertprinzip, und noch viele andere Prinzipien. Versicherungsmathematische Ergebnisse über Prämien betreffen die Eindeutigkeit, die Gleichheit und Vergleichbarkeit, Eigenschaften der Homogeneität, Translationsinvarianz, Iterativität und ihre Charakterisierung, sowie die Erhaltung der Stop-Loss Ordnung und die Vermögensunabhängigkeit.SummaryA new premium principle, the quasi-mean value principle, and a special case thereof, the endogeneous quasi-mean value principle, is introduced. It includes a number of well-known premium principles like the Swiss premium principle, the absolute deviation principle, the Dutch principle, the percentile principle, the Karlsruhe principle, the mean value principle, and many other ones. Actuarial results concern the uniqueness of premiums, equality and comparison of premiums, properties of homogeneity, translation invariance, iterativity and their characterizations, as well as the properties of stop-loss order preservation and wealth independence.
Homogeneous Premium Calculation Principles
Abstract A premium calculation principle π is called positively homogeneous if π(cX) = cπ(X) for all c > 0 and all random variables X. For all known principles it is shown that this condition is fulfilled if it is satisfied for two specific values of c only, say c = 2 and c = 3, and for only all two point random variables X. In the case of the Esscher principle one value of c suffices. In short this means that local homogeneity implies global homogeneity. From this it follows that in the case of the zero utility principle or Swiss premium calculation principle, the underlying utility function is of a very specific type. A very general theorem on premium calculation principles which satisfy a weak continuity condition, is added. Among others the proof uses Kroneckers Theorem on Diophantine Approximations.