We discuss a number of quantile-based risk measures (QBRMs) that have recently been developed in the financial risk and actuarial/insurance literatures. The measures considered include the Value-at-Risk (VaR), coherent risk measures, spectral risk measures, and distortion risk measures. We discuss and compare the properties of these different measures, and point out that the VaR is seriously fl...

This paper introduces a new method for modeling risk averse behavior with spectral risk measures. It is shown that recent approaches, using phenomenological correspondences or results from robust statistics, generally do not generate consistent results. Our method is based on the dual theory of choice. We show that it is possible to encode preference relations in distorted probability measures,...

The paper provides an axiomatic characterization of dynamic risk measures for multi-period financial positions. For the special case of a terminal cash flow, we require that risk depends on its conditional distribution only. We prove a representation theorem for dynamic risk measures and investigate their relation to static risk measures. Two notions of dynamic consistency are proposed. A key i...

In this work we study the Lebesgue property for convex risk measures on the space of bounded càdlàg random processes (R∞). Lebesgue property has been defined for one period convex risk measures in [16] and earlier had been studied in [9] for coherent risk measures. We introduce and study the Lebesgue property for convex risk measures in the multi period framework. We give presentation of all co...

We study the impact of negative returns on the health of a given financial portfolio. It is often the case that a series of significant negative returns trigger a credit event such as a downgrade in rating, or even a default of the portfolio owner. We focus our attention on a Weighted Average of Ordered Returns, which is a statistic that allows us to weight returns according to their relative a...

We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed ...

Nonparametric variance estimation for the distortion risk measure can be readily done through the bootstrap or the nonparametric delta method based on the influence function. The same task for the bootstrapped risk measures, however, has been relatively unexplored in the literature. In this paper we analytically derive the influence function of the exactly bootstrapped quantile and later extend...

With every law invariant coherent risk measure is associated its conditional analogue. In this paper we discuss lower and upper bounds for the corresponding nested (composite) formulations of law invariant coherent risk measures. In particular, we consider the Average Value-at-Risk and comonotonic risk measures. © 2012 Elsevier B.V. All rights reserved.

We consider the problem of representing claims for coherent risk measures. For this purpose we introduce the concept of (predictable and optional ) timeconsistency 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 t...

Properties of distributions are real-valued functionals such as the mean, quantile or conditional value at risk. A property is elicitable if there exists a scoring function such that minimization of the associated risks recovers the property. We extend existing results to characterize the elicitability of properties in a general setting. We further relate elicitability to identifiability (a not...