Paper introduces new risk measures that unify two existing types.
problem Combining two types of risk measures for broader applicability.
method Introduces a new class of risk measures that unify distortion and Haezendonck-Goovaerts measures.
result New risk measures defined on a larger space, with coherent properties in certain scenarios.
Study dynamic risk measures and performance indices using distortion functions.
problem Investigate time consistency of dynamic risk measures and performance indices generated by distortion functions.
method Analyze dynamic coherent risk measures (DCRMs) and dynamic weighted value at risk measures, proving their equivalence. Establish properties of families of DCRMs generated by distortion functions and define corresponding dynamic coherent acceptability indices (DCAIs). Examine time consistency of DCRMs and DCAIs.
result DCRM generated by distortion functions are sub-martingale time consistent but not super-martingale time consistent and not weakly acceptance time consistent.
Establishes a link between risk measures and uniform integrability in finance.
problem Understanding uniform integrability in the context of financial risk measures.
method Introduces the folding score of distortion risk measures to study uniform integrability directly with gains and losses.
result Obtains three sets of equivalent conditions for uniform integrability involving coherent risk measures.
We discuss two distinct approaches, for distorting risk measures of sums of dependent random variables, which preserve the property of coherence. The first, based on distorted expectations, operates on the survival function of the sum. The second, simultaneously applies the distortion on the survival function of the su…
The paper analyzes risk measures and optimal reserve allocation strategies.
problem Risk measures and optimal reserve allocation across multiple lines of business.
method Formalizes expected maximum deficit, introduces implicitly bounded risk measures, and proposes capital allocation approaches.
result Theoretical results on static and dynamic coherence, convexity, and exact optimizations of aggregate minimum reserves.
Submodularity is studied for convex risk measures, including Expected Shortfall.
problem Characterizing submodularity in convex risk measures.
method Analyzing submodularity properties of law-invariant coherent risk measures, including Expected Shortfall and Value-at-Risk.
result AES is submodular only when it reduces to ES, and empirical analysis shows AES violations are less frequent than VaR and ES violations.
Under Solvency II the computation of capital requirements is based on value at risk (V@R). V@R is a quantile-based risk measure and neglects extreme risks in the tail. V@R belongs to the family of distortion risk measures. A serious deficiency of V@R is that firms can hide their total downside risk in corporate network…
Develops a new method for risk diversification using dynamic risk measures.
problem Dynamic risk diversification in investment portfolios.
method Introduces dynamic risk contributions and a recursive optimization approach for coherent dynamic distortion risk measures.
result Dynamic risk budgeting strategies can be solved using deep learning.
Risk-averse model uncertainty framework for safe reinforcement learning.
problem Safe decision making in uncertain environments.
method Risk-averse perspective towards model uncertainty using coherent distortion risk measures; equivalent to distributionally robust safe reinforcement learning problems; efficient, model-free implementation.
result Demonstrates robust performance and safety across perturbed test environments.
Introduces factor risk measures to assess risk relative to multiple factors.
problem Measuring risk relative to multiple factors.
method Introduces a double-argument mapping as a risk measure to assess risk relative to a vector of factors.
result Characterizes various types of factor risk measures including distortion, quantile, linear, and coherent measures.
Efficiently simulates risk budgeting portfolios using novel algorithms.
problem Estimating risk contributions in portfolios efficiently.
method Cutting planes algorithm, specialised SGD for Expected Shortfall, numerical simulations.
result Outperforms standard convex optimisation solvers in estimating risk budgeting portfolios.
New optimal transport divergences derived from scoring functions.
problem Developing new divergences for optimal transport.
method Using scoring functions as cost functions in optimal transport.
result Comonotonic coupling is optimal for many new divergences.
Dual representation and properties of expectile-based expected shortfall studied.
problem Studying the expectile-based expected shortfall as a risk measure.
method Provided dual representation in terms of Bochner integral, showed boundedness properties, and computed for selected distributions.
result Explicit dual representation and boundedness properties of expectile-based expected shortfall.
This paper shows how to calculate risk measures for sums of two counter-monotonic risks.
problem Calculating risk measures for sums of two counter-monotonic risks.
method Using a fixed distortion function and expressing the risk measure of a sum as the sum of two related measures of the marginals.
result The risk measure of a sum of two counter-monotonic risks can be expressed as the sum of two related distortion risk measures of the marginals.
Study distortion risk measures for step-weighted distributions.
problem Analyzing risk measures for specific distribution types.
method Investigate distortion risk measures of step-weighted distributions.
result Developed methods for calculating risk measures.
New method tests risk measures for various distortions.
problem Testing risk measures for different distortions.
method Stratification and randomization of risk levels.
result Method performs well in numerical case studies.
Study on risk measures using distorted Choquet integrals with random distortions.
problem Developing risk measures under random distortions of capacities.
method Introducing and analyzing randomly distorted Choquet integrals with respect to a distorted capacity, establishing properties and providing representations.
result Representation of comonotonic additive conditional risk measures using G-randomly distorted Choquet integrals.
We discuss equivalent axiomatic characterizations of distortion risk measures, and give a novel and concise proof of the characterization of elicitable distortion risk measures. Elicitability has recently been discussed as a desirable criterion for risk measures, motivated by statistical considerations of forecasting. …
Paper proposes efficient method for estimating risk measures in complex models.
problem Accurately estimating distortion risk measures in computationally expensive models.
method Integrates importance sampling and machine learning for efficient Monte Carlo estimation.
result Demonstrates significant reduction in computational cost for estimating risk measures.
Paper proposes a new method to evaluate joint risk under uncertainty.
problem Evaluating joint risk of multiple insurance risks under dependence uncertainty.
method Axiomatic approach to scalar and vector-valued distortion joint risk measures.
result Established a new scalar distortion joint risk measure with positive homogeneity.
The paper explores non-convex risk measures and their characterizations.
problem Characterizing non-convex risk measures without convexity or weak convexity.
method Characterizes monetary risk measures as lower envelopes of families of convex or coherent risk measures, considering law-invariance and SSD-consistency.
result Unified representation theorems for law-invariant risk measures, including VaR.
Paper quantifies distortion risk measures' robustness to distributional uncertainty.
problem Quantifying risk measures' robustness to distributional uncertainty.
method Employing isotonic projections, the paper derives bounds on distortion risk measures' values.
result Sharp bounds on distortion risk measures' values are provided, especially for Value-at-Risk and Range-Value-at-Risk.
Study risk sharing among agents with varying risk preferences.
problem Risk sharing among agents with heterogeneous risk measures.
method Derive explicit solutions for inf-convolution and counter-monotonic inf-convolution under varying risk seeking.
result Explicit solutions for inf-convolution and counter-monotonic inf-convolution can be represented by a generalization of distortion risk measures.
Develops a statistical framework for coherent risk estimation.
problem Constructing coherent risk estimators with sound financial and statistical properties.
method Inspired by axiomatic risk measure theory, defines coherent risk estimators through robust representations linked to L-estimators. result Demonstrates that coherence of a risk measure does not necessarily carry over to its estimators and shows alternative weight structures can lead to different outcomes.
Optimal risk sharing found for heterogeneous risk attitudes using distortion risk measures.
problem Risk sharing in economies with diverse risk attitudes.
method Modeling preferences with distortion risk measures, using comonotonic and counter-monotonic principles.
result Optimal risk sharing strategies identified based on risk attitudes, reducing the n-agent problem to a two-agent formulation. The paper establishes a connection between different risk measures and their risk contributions.
problem Understanding the relationship between conditional coherent and deviation risk measures.
method Axiomatic framework and continuous-time risk contribution analysis.
result Risk contributions of time-consistent risk measures are also time-consistent.
The paper studies robust risk measures with linear penalties under uncertain distributions.
problem Risk measurement under distributional uncertainty.
method Robust distortion risk measures with linear penalty function under distributional constraints.
result Explicit characterization of optimal quantile distribution and value function.
Distortion risk measures are extensively used in finance and insurance applications because of their appealing properties. We present three methods to construct new class of distortion functions and measures. The approach involves the composting methods, the mixing methods and the approach that based on the theory of c…
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 nece…
New method for risk quantification using quantile processes and measure distortions.
problem Risk quantification and valuation in financial markets.
method Develops a novel stochastic valuation principle based on probability measure distortions induced by quantile processes.
result Introduces a system of subjective probability measures that indexes a stochastic valuation principle susceptible to probability measure distortions.
In this paper, we introduce the rich classes of conditional distortion (CoD) risk measures and distortion risk contribution (ΔCoD) measures as measures of systemic risk and analyze their properties and representations. The classes include the well-known conditional Value-at-Risk, conditional Expected Shortfall, and r…
Paper proposes robust risk measures for non-negative risks with partial information.
problem Tackles robustness of distortion risk measures under distributional uncertainty.
method Introduces new uncertainty sets and derives closed-form expressions for risk maximization.
result Derives closed-form expressions for risk maximization over uncertainty sets.
Study dynamic Pareto-optimal allocations in multi-period economies with time-consistent risk measures.
problem Optimal allocation in multi-period pure-exchange economies with stochastic endowments and time-consistent risk measures.
method Introduced dynamic Pareto-optimal allocation processes and derived recursive and comonotone improvement theorems.
result Dynamic Pareto-optimal allocation processes can be constructed recursively and are comonotone.
The paper analyzes worst-case distortion risk metrics and weighted entropy under partial information.
problem Analyzing worst-case distortion risk metrics and weighted entropy with limited information.
method General distributions, partial information (mean and variance), various entropies and risk measures.
result Provides worst-case results for distortion risk metrics and weighted entropy.
Paper derives best- and worst-case GlueVaR measures with incomplete data.
problem Risk measurement with limited information and shape constraints.
method Unified framework based on partial distribution information and shape properties.
result Characterization of extremal GlueVaR distributions with convex envelopes.
The paper analyzes extreme risk measures with limited distributional information.
problem Investigating risk measures under partial knowledge of distribution moments and shape.
method Employing probability inequalities and modified Schwarz inequality to derive bounds on distortion risk measures.
result Unified framework for calculating best- and worst-case scenarios of distortion risk measures.
In this paper we present a theoretical framework for studying coherent acceptability indices in a dynamic setup. We study dynamic coherent acceptability indices and dynamic coherent risk measures, and we establish a duality between them. We derive a representation theorem for dynamic coherent risk measures in terms of …
Constructs new elicitable risk measures with multiplicative scoring functions.
problem Defining new elicitable risk measures with specific properties.
method Constructs new elicitable risk measures using a multiplicative scoring function.
result Encompasses and allows construction of novel elicitable risk measures.
Paper introduces new risk norms based on ES with flexible distortion functions.
problem Risk quantification and anomaly detection in financial data.
method Developed generalized Expected-Shortfall (ES) norms using distortion risk measures and duality theory.
result Unified analytical framework for risk quantification and practical applications.
Develops non-standard analysis for coherent risk estimation.
problem Estimating coherent risk measures in financial contexts.
method Non-standard analysis, hyperfinite representations, discrete Kusuoka formulae, plug-in asymptotics.
result Uniform almost sure consistency and asymptotic normality of spectral plug-in estimators.
We give a complete characterization of both comonotone and not comonotone coherent risk measures in the discrete finite probability space, where each outcome is equally likely. To the best of our knowledge, this is the first work that characterizes \textit{and} distinguishes comonotone and not comonotone coherent risk …
Paper introduces DCoVaR for aggregate risk models, outperforming existing methods.
problem Lack of coherent risk measures for aggregate risk models.
method Proposes Dependent Conditional Value-at-Risk (DCoVaR) for a target loss dependent on another random loss.
result DCoVaR outperforms MCoVaR and CCoVaR in numerical simulations and empirical studies.
In this paper we introduce a new coherent cumulative risk measure on RLp, the space of càdlàg processes having Laplace transform. This new coherent risk measure turns out to be tractable enough within a class of models where the aggregate claims is driven by a spectrally positive Lévy process. Moreover, w…
We propose a pricing technique based on coherent risk measures, which enables one to get finer price intervals than in the No Good Deals pricing. The main idea consists in splitting a liability into several parts and selling these parts to different agents. The technique is closely connected with the convolution of coh…
We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks: - coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath; - factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price …
Study optimal risk sharing in decentralized peer-to-peer markets with robust risk measures.
problem Optimizing risk sharing in decentralized markets with non-convex risk measures.
method Characterization of Pareto-optimal allocations using robust distortion risk measures and probabilistic risk aversion.
result Shape of allocations depends on agents' tail risk assessments.
The problem behind this paper is the proper measurement of the degree of quality/acceptability/distance to arbitrage of trades. We are narrowing the class of coherent acceptability indices introduced by Cherny and Madan (2007) by imposing an additional mathematical property. For this, we introduce the notion of a conca…
It is shown that the axioms for coherent risk measures imply that whenever there is an asset in a portfolio that dominates the others in a given sample (which happens with finite probability even for large samples), then this portfolio cannot be optimized under any coherent measure on that sample, and the risk measure …