## On a risk measure related to the certainty equivalent under Cumulative Prospect Theory

Jacek Chudziak

University of Rzeszow, Department of Mathematics, Al. Rejtana 16A, Rzeszów 35-959, Poland

 Abstract The Cumulative Prospect Theory was created in 1992 by A. Tversky and D. Kahneman. It is based on several experiments carried out by them. The experiments showed that, making decisions under risk, people set a reference point and consider the lower outcomes as losses and larger ones as gains. Furthermore, people distort probabilities. In general, the probabilities of gains and losses are distorted in a different way. Under Cumulative Prospect Theory, the preference relation of the decision maker over a given family of lotteries Δ is represented by the functional V:Δ-->R of the form                                                                                               V(X)=Egh u(X),                                     (1) where Egh is the generalized Choquet integral with respect to the probability distortion functions g and h for gains and losses, respectively, and u:R-->R is an increasing and continuous function with u(0)=0, called a value function. From the properties of the generalized Choquet integral and a value function it follows that for every X from Δ there is a unique real number C(X) such that                                                 V(C(X))=V(X), where C(X) denotes a constant random variable, identically equal to C(X). A functional C:Δ-->R defined in this way is called the certainty equivalent. In view of (1), we have                                               C(X)=u-1(Egh u(X)). A functional ρ:= -C can be treated a risk measure. In the talk we deal with the properties of the functional ρ.

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