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.
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 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. …
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.
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.
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.
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. 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 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…
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.
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.
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.
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.
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.
In this paper, we consider the problem of optimal reinsurance design, when the risk is measured by a distortion risk measure and the premium is given by a distortion risk premium. First, we show how the optimal reinsurance design for the ceding company, the reinsurance company and the social planner can be formulated i…
Develops a framework for robust RL with dynamic risk measures.
problem Optimal RL strategies depend on risk preferences and model dynamics.
method Dynamic robust distortion risk measures, Wasserstein ball, neural networks, strictly consistent scoring functions, policy gradient formulae, actor-critic algorithm.
result Demonstrates improved performance in portfolio allocation example.
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…
The paper studies risk-sharing allocations for risk-seeking agents using a common distortion risk measure.
problem Characterizing Pareto-optimal risk-sharing allocations for risk-seeking agents.
method Modeling preferences with a common distortion risk measure and analyzing three settings: risk-averse, risk-seeking, and inverse S-shaped distortion.
result Pareto-optimal allocations for risk-seeking agents are counter-monotonic, not comonotonic.
Optimizes riskmetrics with uncertainty, making complex problems simpler.
problem Optimizing riskmetrics with distributional uncertainty.
method Unifying result converting non-convex optimization to convex, using closedness under concentration.
result Great tractability achieved through unifying equivalence result.
The paper addresses risk sharing and variability measures among agents with general risk preferences.
problem Risk sharing and variability measures among agents with general risk preferences.
method Characterizes Pareto-optimal allocations using Gini deviation, mean-median deviation, and inter-quantile difference as variability measures.
result Optimal allocations are not comonotonic and feature a mixture of pairwise counter-monotonic structures.
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.
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.
We characterize when a convex risk measure associated to a law-invariant acceptance set in L∞ can be extended to Lp, 1≤p<∞, preserving finiteness and continuity. This problem is strongly connected to the statistical robustness of the corresponding risk measures. Special attention is paid to concre…
Investors optimize their portfolios within a Wasserstein ball to match a benchmark's risk profile.
problem Optimizing portfolio performance while maintaining risk proximity to a benchmark.
method Optimal dynamic strategy selection based on minimizing distortion risk measures within a Wasserstein ball.
result An optimal dynamic strategy exists and can be calculated through isotonic projections.
The paper calculates bounds for risk metrics and entropies under partial information constraints.
problem Analyzing risk metrics and entropies for unimodal, symmetric distributions with limited information.
method Develops lower and upper bounds for worst-case distortion riskmetrics and weighted entropy for unimodal, symmetric distributions with known mean and variance.
result Sharp upper bounds for distortion riskmetrics and weighted entropy for symmetric distributions.
The aims of this study are twofold. First, we consider an optimal risk allocation problem with non-convex preferences. By establishing an infimal representation for distortion risk measures, we give some necessary and sufficient conditions for the existence of optimal and asymptotic optimal allocations. We will show th…
The paper examines how risk reduction and insurance choices interact under convex premium principles.
problem Interaction between self-protection and insurance demand under convex premium principles.
method Investigates optimal prevention efforts and insurance shares using distortion risk measures.
result Self-protection and insurance are complementary, but ex ante moral hazard can turn this into a substitution effect.
Study risk-sensitive reinforcement learning with Lipschitz dynamic risk measures, establishing regret bounds.
problem Risk-sensitive reinforcement learning in Markov decision processes.
method Two model-based algorithms for Lipschitz dynamic risk measures, focusing on regret bounds.
result Upper bounds demonstrate optimal dependencies on actions and episodes, reflecting risk sensitivity vs. sample complexity trade-off.
Method determines asset prices in incomplete markets to optimize portfolios.
problem Optimizing portfolios in incomplete markets with price constraints.
method Maximum entropy in the mean to adjust distortion function from bid-ask data.
result Prices of assets comply with portfolio optimization constraints.
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.
Sharp bounds for distortion risk metrics under uncertain distributions.
problem Modeling risk metrics under distributional uncertainty.
method Established bounds for distortion risk metrics using specific features of underlying distributions.
result Identified worst- and best-case values of distortion risk metrics.
Distortion (Denneberg 1990) is a well known premium calculation principle for insurance contracts. In this paper, we study sensitivity properties of distortion functionals w.r.t. the assumptions for risk aversion as well as robustness w.r.t. ambiguity of the loss distribution. Ambiguity is measured by the Wasserstein d…
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.
In this paper, we study two classes of optimal reinsurance models from perspectives of both insurers and reinsurers by minimizing their convex combination where the risk is measured by a distortion risk measure and the premium is given by a distortion premium principle. Firstly, we show that how optimal reinsurance mod…
This paper studies the problem of optimal investment with CRRA (constant, relative risk aversion) preferences, subject to dynamic risk constraints on trading strategies. The market model considered is continuous in time and incomplete. the prices of financial assets are modeled by Itô processes. The dynamic risk constr…
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.
Paper introduces new risk measures for default risk and model uncertainty.
problem Model uncertainty and default risk in rating systems.
method Introduces default risk measures and discusses their properties and impacts.
result Different default risk measures and margins of conservatism affect risk-weighted assets.
Study asymptotic properties of generalized shortfall risk measures for heavy-tailed risks.
problem Understanding risk measures for heavy-tailed risks.
method Derive asymptotic expansions for generalized shortfall risk measures.
result Unified theory for risk measures including distortion and utility-based measures.