An Iterativity Condition for the Mean-Value Principle Under Cumulative Prospect Theory

Abstract

In this paper, we present the full characterization of the iterativity condition for the mean-value principle under the cumulative prospect theory. It turns out that the premium principle is iterative for exactly six pairs of probability distortion functions. Some of the corresponding premium principles are the classical mean-value principle, essential infimum or essential supremum of the random loss. Moreover, from the proof of the main theorem of this paper, it follows that the iterativity of the mean-value principle is equivalent to the iterativity of the generalized Choquet integral.

Disappointment Aversion Premium Principle

“…Our results demonstrate that equivalent utility premium principle under the DA model is more realistic than that under the EUT, in the sense that the premium principle representations under the linear and exponential utilities unveil an interesting connection with the capital reserve regulatory requirement in practice. Our work complements Tsanakas and Desli (2003) and Kaluszka and Krzeszowiec (2012), who are the first laying down the zero-utility premium principle under behavioral frameworks.…”

“…7is that H is the minimal possible price to be charged so that the pricing agent with an initial wealth w is not worse off by undertaking the financial obligation. This actuarial indifference determination for the insurance premium has been thoroughly studied in the recent decades, see, for instance, Denuit et al (2006), Heilpern (2003), Kaluszka and Krzeszowiec (2012), Laeven and Goovaerts (2008), Tsanakas andDesli (2003, 2005) and Tsanakas (2008). In the field of finance, the indifference price is defined through a more general equation, which allows the pricing agent to invest the initial fortune and the reward arising from transaction if any, in order to fulfill the future financial obligation.…”

“…Tsanakas and Desli (2003) derived a premium principle called the generalized expected utility premium principle based on the rank-dependent EUT developed by Quiggin (1993); in particular, their proposed premium principle is a combination of the utility function of X and the distortion on the loss survival distribution. Besides, Kaluszka and Krzeszowiec (2012) studied the equivalent utility premium principle based on the Cumulative Prospect Theory developed by Kahneman and Tversky (1992).…”

On Jensen's inequality for generalized Choquet integral with an application to risk aversion

“…Since the function u is strictly increasing and continuous, the premium π u (X, w) exists and is determined uniquely if w − X ∈ X u , where X u denotes the set of such measurable functions X : Ω → R that C µν (X) ∈ I and C µν (u(X)) ∈ u(I). The premium π u was proposed in [20,21] in case of the capacities being Kahneman-Tverski distorted measures (see the Appendix, Example 2). A lot of properties of that premium was studied, but the problem of risk aversion measure was not examined.…”

“…It is worthy to emphasize that Kahneman was awarded with Nobel Prize in Economic Sciences in 2002 for that theory. The capacities also became a basic tool to measure risk in insurance mathematics [9,20,21].…”