Synergy Modelling and Financial Valuation : the contribution of Fuzzy Integrals.

Les méthodes d’évaluation financière utilisent des opérateurs d’agrégation reposant sur les propriétés d’additivité (sommations, intégrales de Lebesgue). De ce fait, elles occultent les phénomènes de renforcement et de synergie (ou de redondance) qui peuvent exister entre les éléments d’un ensemble organisé. C’est particulièrement le cas en ce qui concerne le problème d’évaluation financière du patrimoine d’une entreprise : en effet, en pratique, il est souvent mis en évidence une importante différence de valorisation entre l’approche « valeur de la somme des éléments » (privilégiant le point de vue financier) et l’approche « somme de la valeur des différents éléments » (privilégiant le point de vue comptable). Les possibilités offertes par des opérateurs d’agrégation comme les intégrales floues (Sugeno, Grabisch, Choquet) permettent, au plan théorique, de modéliser l’effet de synergie. La présente étude se propose de valider empiriquement les modalités d’implémentation opérationnelle de ce modèle à partir d’un échantillon d’entreprises cotées ayant fait l’objet d’une évaluation lors d’une OPA.Financial valuation methods use additive aggregation operators. But a patrimony should be regarded as an organized set, and additivity makes it impossible for these aggregation operators to formalize such phenomena as synergy or mutual inhibition between the patrimony’s components. This paper considers the application of fuzzy measure and fuzzy integrals (Sugeno, Grabisch, Choquet) to financial valuation. More specifically, we show how integration with respect to a non additive measure can be used to handle positive or negative synergy in value construction.Fuzzy measure; Fuzzy integral; Aggregation operator; Synergy; Financial valuation;

Quasi-arithmetic means and OWA functions in interval-valued and Atanassov's intuitionistic fuzzy set theory

In this paper we propose an extension of the well-known OWA functions introduced by Yager to interval-valued (IVFS) and Atanassov’s intuitionistic (AIFS) fuzzy set theory. We first extend the arithmetic and the quasi-arithmetic mean using the arithmetic operators in IVFS and AIFS theory and investigate under which conditions these means are idempotent. Since on the unit interval the construction of the OWA function involves reordering the input values, we propose a way of transforming the input values in IVFS and AIFS theory to a new list of input values which are now ordered

On Lipschitz properties of generated aggregation functions

This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element e &isin;]0, 1[.<br /

Construction of k-Lipschitz triangular norms and conorms from empirical data

This paper examines the practical construction of k-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on k-convex additive generators and translate k-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees k-Lipschitz property of the resulting triangular norms and conorms.<br /

The quest for rings on bipolar scales

We consider the interval $]{-1},1[$ and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t.~$0$ are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.

Some results on Lipschitz quasi-arithmetic means

We present in this paper some properties of k-Lipschitz quasi-arithmetic means. The Lipschitz aggregation operations are stable with respect to input inaccuracies, what is a very important property for applications. Moreover, we provide sufficient conditions to determine when a quasi&ndash;arithemetic mean holds the k-Lipschitz property and allow us to calculate the Lipschitz constant k.<br /

Basic generated universal fuzzy measures

AbstractThe concept of basic generated universal fuzzy measures is introduced. Special classes and properties of basic generated universal fuzzy measures are discussed, especially the additive, the symmetric and the maxitive case. Additive (symmetric) basic universal fuzzy measures are shown to correspond to the Yager quantifier-based approach to additive (symmetric) fuzzy measures. The corresponding fuzzy integral-based aggregation operators are introduced, including the generated OWA operators

Constructions of aggregation operators that preserve ordering of the data

We address the issue of identifying various classes of aggregation operators from empirical data, which also preserves the ordering of the outputs. It is argued that the ordering of the outputs is more important than the numerical values, however the usual data fitting methods are only concerned with fitting the values. We will formulate preservation of the ordering problem as a standard mathematical programming problem, solved by standard numerical methods.<br /