Representation of concave distortions and applications

ABSTRACT

A family of concave distortion functions is a set of concave and increasing functions, mapping the unity interval onto itself. Distortion functions play an important role defining coherent risk measures. We prove that any family of distortion functions which fulfils a certain translation equation, can be represented by a distribution function. An application can be found in actuarial science: moment-based premium principles are easy to understand but in general are not monotone and cannot be used to compare the riskiness of different insurance contracts with each other. Our representation theorem makes it possible to compare two insurance risks with each other consistent with a moment-based premium principle by defining an appropriate coherent risk measure.

KEYWORDS:

• Representation of distortion functions
• premium principle
• coherent risk measure
• WANG-transform
• log-concavity

JEL CLASSIFICATIONS:

• C00
• G22

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

1 For example let X take the values 10 or 90, each with the same probability. Clearly, X is less risky than the constant Z=100. Let $\gamma =1$. The standard deviation premium of X is about 106 but the premium of Z is smaller, it is equal to 100.

Additional information

Funding

This research is partially supported by 13th UAB-PIF scholarship. The funding source has no involvement in the writing of this article.