Some new classes of consistent risk measures

2015

ABSTRACT: In actuarial literature the properties of risk measures or insurance premium prin-ciples have been extensively studied. We propose a characterization of a particular class of coherent risk measures. The considered premium principles are obtained by expansion of TVar measures.

Mathematical and Statistical Methods in Insurance and Finance, 2008

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied . We propose a characterization of a particular class of coherent risk measures . The considered premium principles are obtained by expansion of TVar measures.

Mathematical Finance, 2007

The optimized certainty equivalent (OCE) is a decision theoretic criterion based on a utility function, that was first introduced by the authors in 1986. This paper re-examines this fundamental concept, studies and extends its main properties, and put it in perspective to recent concepts of risk measures. We show that the negative of the OCE naturally provides a wide family of risk measures that fits the axiomatic formalism of convex risk measures. Duality theory is used to reveal the link between the OCE and the ϕ-divergence functional (a generalization of relative entropy), and allows for deriving various variational formulas for risk measures. Within this interpretation of the OCE, we prove that several risk measures recently analyzed and proposed in the literature (e.g., conditional value of risk, bounded shortfall risk) can be derived as special cases of the OCE by using particular utility functions. We further study the relations between the OCE and other certainty equivalents, providing general conditions under which these can be viewed as coherent/convex risk measures. Throughout the paper several examples illustrate the flexibility and adequacy of the OCE for building risk measures.

services.economia.unitn.it

Insurance: Mathematics and Economics, 2022

In this paper, we show that, on classical model spaces including Orlicz spaces, every real-valued, law-invariant, coherent risk measure automatically has the Fatou property at every point whose negative part has a thin tail.

2015

This thesis deals with the risk theory in Finance and Insurance. Application of the Comonotonicity concept, the strongest risk dependence, is described for identifying the Pareto optima and Individually Rational Pareto optima allocations, option pricing and quantification of risk. Furthermore it is shown that the left monotone risk aversion, a meaningful refinement of strong risk aversion, characterizes Yaari’s decision makers for whom deductible insurance is optimal. The concept of Comonotonicity is introduced and discussed in Chapter 1. In case of multiple risks, the idea that a natural way for insurance companies to optimally share risks is risk by risk Pareto-optimality is adopted. Moreover, the Pareto optimal and individually Pareto optimal allocations are characterized. The Chapter 2 investigates the application of the Comonotonicity concept in option pricing and quantification of risk. A novel control variate Monte Carlo method is introduced and its application is explained fo...

We consider the problem of representing claims for coherent risk measures. For this purpose we introduce the concept of (predictable and optional ) time-consistency with respect to a portfolio of assets, generalizing the one defined by Delbaen. In a similar way we extend the notion of multiplicative, or m-stability, by introducing predictable and optional versions for a portfolio of assets. We then prove that the two concepts of m-stability and time-consistency are equivalent, thus giving necessary and sufficient conditions for a coherent risk measure to be represented by a market with proportional transaction costs. Finally we show that any market with proportional transaction costs is equivalent to a market priced by a coherent risk measure, essentially establishing the equivalence of the two concepts.

Coherent risk measures have lately become a hot topic in both theoretical research and practical applications. Since they possess a number of disadvantages, some generalizations and modifications have been proposed recently. In the present paper we propose one more generalization which holds a number of attractive properties. A representation theorem for the risk measures class has been obtained, properties of functionals have been established, partial cases and examples are also presented.

Mathematical Finance, 2002

This note defines the premium of a put option on the firm as a measure of insolvency risk. The put premium is not a coherent risk measure as defined by Artzner et al. (1999). It satisfies all the axioms for a coherent risk measure except one, the translation invariance axiom. However, it satisfies a weakened version of the translation invariance axiom that we label translation monotonicity. The put premium risk measure generates an acceptance set that satisfies the regularity Axioms 2.1-2.4 of Artzner et al. (1999). In fact, this is a general result for any risk measure satisfying the same risk measure axioms as the put premium. Finally, the coherent risk measure generated by the put premium's acceptance set is the minimal capital required to protect the firm against insolvency uniformly across all states of nature.

ESAIM: Proceedings, 2014

The goal of this paper is to give recent results in risk theory presented at the Conference "Journée MAS 2012" which took place in Clermont Ferrand. After a brief state of the art on ruin theory, we explore some particular aspects and recent results. One presents matrix exponential approximations of the ruin probability. Then we present asymptotics of the ruin probability based on mixing properties of the claims distribution. Finally, the multivariate case, motivated by reinsurance, is presented and some contemporary results (closed forms and asymptotics) are given.