Coherent Risk Measures and Upper Previsions

Risk and Insurance, 2003

In this paper we introduce convex imprecise previsions as a special class of imprecise previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise previsions but are not necessarily positively homogeneous. The broader class of weakly convex imprecise previsions is also studied and its fundamental properties are demonstrated. The notions of weak convexity and convexity are then applied to risk measurement, leading to a more general definition of convex risk measure than the one already known in risk measurement literature.

SSRN Electronic Journal, 2021

A risk analyst assesses potential financial losses based on multiple sources of information. Often, the assessment does not only depend on the specification of the loss random variable, but also various economic scenarios. Motivated by this observation, we design a unified axiomatic framework for risk evaluation principles which quantifies jointly a loss random variable and a set of plausible probabilities. We call such an evaluation principle a generalized risk measure. We present a series of relevant theoretical results. The worst-case, coherent, and robust generalized risk measures are characterized via different sets of intuitive axioms. We establish the equivalence between a few natural forms of law invariance in our framework, and the technical subtlety therein reveals a sharp contrast between our framework and the traditional one. Moreover, coherence and strong law invariance are derived from a combination of other conditions, which provides additional support for coherent risk measures such as Expected Shortfall over Valueat-Risk, a relevant issue for risk management practice.

2001

We study a space of coherent risk measures M_phi obtained as certain expansions of coherent elementary basis measures. In this space, the concept of ``Risk Aversion Function'' phi naturally arises as the spectral representation of each risk measure in a space of functions of confidence level probabilities. We give necessary and sufficient conditions on phi for M_phi to be a coherent measure. We find in this way a simple interpretation of the concept of coherence and a way to map any rational investor's subjective risk aversion onto a coherent measure and vice--versa. We also provide for these measures their discrete versions M_phi^N acting on finite sets of N independent realizations of a r.v. which are not only shown to be coherent measures for any fixed N, but also consistent estimators of M_phi for large N. Finally, we find in our results some interesting and not yet fully investigated relationships with certain results known in insurance mathematical literature.

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.

Preprint, http://www2. warwick. ac. uk/fac/ …, 2008

Abstract. We consider the problem of representing claims for coherent risk mea-sures. 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 ...

New Dimensions in Fuzzy Logic and …, 2007

Because of their simplicity, risk measures are often employed in financial risk evaluations and related decisions. In fact, the risk measure ρ(X) of a random variable X is a real number customarily determining the amount of money needed to face the potential losses X might cause. At a sort of second-order level, the adequacy of ρ(X) may be investigated considering the part of the losses it does not cover (its shortfall). This may suggest employing a further, more prudential risk measure, taking the shortfall of ρ(X) into account. In this paper a family of shortfalldependant risk measures is proposed, investigating its consistency properties and its utilization in insurance pricing. These results are obtained and subsequently extended within the framework of imprecise previsions, of which risk measures are an instance. This also leads us to investigate properties of a rather weak consistency notion for imprecise previsions, termed 1–convexity.

Insurance: Mathematics and Economics, 2004

Many types of insurance premium principles and/or risk measures can be characterized by means of a set of axioms which in many cases are arbitrarily chosen and not always in accordance with economic reality. In the present paper we generalize Yaari's risk measure based on some less stringent axioms. In addition we derive translation invariant minimal Orlicz risk measures, which we call Haezendonck risk measures, and obtain sufficient conditions on the risk measure of Bernoulli risks to fulfill additivity and superadditivity properties for Orlicz premium principles. Keywords: Consistent risk measures, Haezendonck risk measure, Yaari's dual theory of choice under risks 1 Introduction Recently, in Goovaerts et al. (2003 a ) it was argued that risk measures should be selected in an appropriate way in order to reflect the basic economic underlying reality. Indeed several examples can be given, which are relevant to real life insurance problems where evidently the properties that the risk measures should have are determined by the realities of the actuarial applications.

Journal of Banking & Finance, 2007

Coherent measures of risk defined by the axioms of monotonicity, subadditivity, positive homogeneity, and translation invariance are recent tools in risk management to assess the amount of risk agents are exposed to. If they also satisfy law invariance and comonotonic additivity, then we get a subclass of them: spectral measures of risk. Expected shortfall is a well-known spectral measure of risk.

2007

Extending the approach of Jouini et al. we define set-valued (convex) measures of risk and its acceptance sets. Using a new duality theory for set-valued convex functions we give dual representation theorems. A scalarization concept is introduced that has economical meaning in terms of prices of portfolios of reference instruments. Using primal and dual descriptions, we introduce new examples for set-valued measures of risk, e.g. set-valued expectations, Value at Risk, Average Value at Risk and entropic risk measure.

2017

This study is about developing some further ideas in imprecise probability models of financial risk measures. A financial risk measure has been interpreted as an upper prevision of imprecise probability, which through the conjugacy relationship can be seen as a lower prevision. The risk measures selected in the study are value-at-risk (V aR) and conditional value-at-risk (CV aR). The notion of coherence of risk measures is explained. Stocks that are traded in the financial markets (the risky assets) are seen as the gambles. The study makes a determination through computation from actual assets data whether the risk measure assessments of gambles (assets) are coherent as an imprecise probability. It is observed that coherence of assessments depends on the asset’s returns distribution characteristic.