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A Remark on the Principle of Zero Utility

Published online by Cambridge University Press:  29 August 2014

Hans U. Gerber
Hans U. Gerber*
Affiliation:
University of Lausanne, Switzerland
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Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equation

which is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense that

for an arbitrarily chosen point y. Alternatively, one can consider the risk aversion

which is the same for all affine transformations of a utility function.

Given the risk aversion r(x), the standardized utility function can be retrieved from the formula

It is easily verified that this expression satisfies (2) and (3).

The following lemma states that the greater the risk aversion the greater the premium, a result that does not surprise.

Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 13 , Issue 2 , December 1982 , pp. 133 - 134
Copyright
Copyright © International Actuarial Association 1982

References

REFERENCES

Gerber, H. U. (1974) On Additive Premium Calculation Principles. Astin Bulletin 7, 215–222.CrossRefGoogle Scholar
Leepin, P. (1975) Ueber die Wahl von Nutzenfunktionen für die Bestimmung von Versicherungsprämien. Mitteilungen der Vereinigung schweizerischer Versicherunsmathematiker 75, 27–45.Google Scholar
Pratt, J. W. (1964). Risk Aversion in the Small and in the Large. Econometrica 32, 122–136.CrossRefGoogle Scholar