This paper proves Expected Shortfall is concave, not convex.
problem Understanding the convexity/concavity of Expected Shortfall.
method Analytical proof of concavity with respect to probability distributions.
result Expected Shortfall is concave, not convex.
Simplifies study of multivariate shortfall risk measures.
problem Complexity in studying multivariate shortfall risk measures.
method Defines shortfall risk measures through a 1-dimensional function.
result Simplifies properties of multivariate shortfall risk measures.
We refine Expected Shortfall by controlling different tail portions, offering tailored risk assessments.
problem Risk assessment in financial positions, especially in tail regions.
method Introducing adjusted Expected Shortfall measures that control different tail portions.
result Adjusted Expected Shortfall measures ensure risk does not exceed specified thresholds for various probability levels.
This paper introduces new risk measures for evaluating losses with varying time horizons.
problem Capturing horizon risk and cash non-additivity in risk evaluation.
method Uses BSDEs and shortfall approaches to develop h-generalized shortfall risk measures.
result Introduces hq-entropic risk measures as a new family of fully-dynamic risk measures.
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.
Study on expectile and expected shortfall for tail risk assessment.
problem Comparing expectile and expected shortfall for tail risk assessment.
method Duality results and optimized certainty equivalent.
result Derived bounds and asymptotic behavior of expectile with respect to expected shortfall.
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.
Investigates risk measures for DC pension decumulation.
problem Develop optimal decumulation strategies for DC plan holders.
method Formulates decumulation as a control problem, studies risk measures (expected shortfall, linear shortfall, probability of shortfall).
result Optimal controls for expected reward and expected shortfall are identical to those for expected reward and linear shortfall.
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.
Investigates a new measure PELVE_n for risk assessment.
problem Estimating higher-order risk measures in finance.
method Mathematical analysis and distribution-specific calculations.
result Developed and analyzed PELVE_n for various distributions.
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.
Financial institutions have to allocate so-called "economic capital" in order to guarantee solvency to their clients and counter parties. Mathematically speaking, any methodology of allocating capital is a "risk measure", i.e. a function mapping random variables to the real numbers. Nowadays "value-at-risk", which is d…
New AI models improve financial hedging by reducing shortfall and tail risk.
problem Static model calibration gaps in derivatives markets.
method Two reinforcement learning frameworks: RLOP and QLBS.
result RLOP reduces shortfall frequency and improves tail risk in stress scenarios.
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.
Introduces Lambda Expected Shortfall as a risk measure generalizing ES.
problem Lack of a comprehensive risk measure that generalizes ES and Lambda-VaR.
method Introduces Lambda-ES, a new risk measure with explicit formula and properties.
result Lambda-ES is the smallest quasi-convex and law-invariant risk measure dominating Lambda-VaR.
Study shows equivalence of four risk constraints in non-concave optimization problems.
problem Investigating risk constraints in non-concave optimization for financial companies.
method Analytical solutions for four risk constraints (ES, EDS, VaR, AVaR) under non-concave optimization.
result All four risk constraints lead to the same optimal solution, differing from concave optimization.
Expected Shortfall (ES) in several variants has been proposed as remedy for the defi-ciencies of Value-at-Risk (VaR) which in general is not a coherent risk measure. In fact, most definitions of ES lead to the same results when applied to continuous loss distributions. Differences may appear when the underlying loss di…
Study improves accuracy of risk measures using advanced algorithms.
problem Computing accurate risk measures for financial losses.
method Nested stochastic approximation and multilevel acceleration.
result Established central limit theorems for estimation errors.
Study on optimal strategies for minimizing shortfall risk in game options.
problem Existence of optimal hedging strategies for shortfall risk in game options.
method Continuous time Black--Scholes model, finite and infinite exercise times.
result Optimal strategies exist for finite exercise times but not for all time intervals.
A new tail-shape index based on Value at Risk and Expected Shortfall.
problem Measuring and comparing tail behavior of loss distributions.
method Introducing a new θ-index based on equal level relationships between Value at Risk and Expected Shortfall. result The θ-index provides a level-dependent, scale-free measure of upper tail behavior. We present the Shortfall Deviation Risk (SDR), a risk measure that represents the expected loss that occurs with certain probability penalized by the dispersion of results that are worse than such an expectation. SDR combines Expected Shortfall (ES) and Shortfall Deviation (SD), which we also introduce, contemplating t…
New risk measures adjust for tail risk inadequacies.
problem Tail risk inadequacy in classical risk measures.
method Developed a family of adjusted risk measures using target risk profiles.
result Analyzed and derived properties of adjusted risk measures.
Quantum algorithm for dynamic asset allocation using expected shortfall.
problem Dynamic risk management in finance, especially tail risks.
method Quantum annealing algorithm in QUBO form for expected shortfall constraint.
result Quantum algorithm provides a faster solution for dynamic asset allocation.
We study shortfall risk minimization for American options with path dependent payoffs under proportional transaction costs in the Black--Scholes (BS) model. We show that for this case the shortfall risk is a limit of similar terms in an appropriate sequence of binomial models. We also prove that in the continuous time …
Ineffective risk measures fail to control risky investor behavior in markets with arbitrage opportunities.
problem Ineffectiveness of coherent risk measures in managing risky investor behavior in markets with arbitrage opportunities.
method Analytical determination of ρ-arbitrage portfolios and consideration of realistic numerical examples of incomplete markets. result Expected shortfall constraints can be ineffective in realistic markets, but reasonable expected utility constraints are effective.
Combines VaR and ES forecasts for cryptocurrency market risk management.
problem Improving tail risk forecasts in financial markets.
method Proposes semiparametric and parametric combination frameworks.
result Combined forecasts outperform individual VaR and ES forecasts.
New method optimizes risk estimation for financial losses.
problem Estimating expected shortfall risk for heavy-tailed distributions.
method Proposes a novel estimator for expected shortfall robust to data corruption.
result Demonstrates superior performance and robustness compared to classical methods.
New methods estimate multivariate shortfall risk more efficiently.
problem Estimating multivariate shortfall risk is computationally challenging.
method Combines Fourier inversion and RQMC sampling in frequency domain.
result Fourier RQMC methods outperform existing benchmarks.
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.
In this paper, we generalize the parametric delta-VaR method from portfolios with normally distributed risk factors to portfolios with elliptically distributed ones. We treat both the expected shortfall and the Value-at-Risk of such portfolios. Special attention is given to the particular case of a multivariate t-distr…
CAESar improves risk forecasting by combining VaR and ES estimates.
problem Lack of tail risk measures in financial risk management.
method Conditional Autoregressive Expected Shortfall model, combining VaR and ES estimates.
result CAESar outperforms existing methods in risk forecasting.
This paper presents analytical solutions to the problem of how to calculate sensible VaR (Value-at-Risk) and ES (Expected Shortfall) contributions in the CreditRisk+ methodology. Via the ES contributions, ES itself can be exactly computed in finitely many steps. The methods are illustrated by numerical examples.
New approximations for Value at Risk and Expected Shortfall accounting for kurtosis.
problem Approximating Value at Risk and Expected Shortfall with positive skewness and kurtosis.
method Extensions of the Normal Power Approximation incorporating skewness and kurtosis.
result Improved precision for various loss distributions.
We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black--Scholes market considering Lipschitz continuous path-dependent payoffs for both discrete- and continuous-time cases. These results are new also for usual American style option…
We consider the pricing and hedging of exotic options in a model-independent set-up using \emph{shortfall risk and quantiles}. We assume that the marginal distributions at certain times are given. This is tantamount to calibrating the model to call options with discrete set of maturities but a continuum of strikes. In …
Optimizes shortfall risk using gradient-based methods.
problem Optimizing utility-based shortfall risk measures.
method Gradient-based stochastic optimization, non-asymptotic bounds derivation.
result Non-asymptotic convergence rate for optimizing UBSR.
Optimal retirement timing and consumption under shortfall risk management
problem Optimal portfolio, consumption, and endogenous early retirement problem
method Maximizing expected lifetime consumption utility while managing the maximum wealth shortfall relative to a benchmark
result Geometric structure of the stopping set and feedback-form optimal retirement boundary
The problem of estimation error of Expected Shortfall is analyzed, with a view of its introduction as a global regulatory risk measure.
New tests for ES risk measure introduced to assess forecast quality.
problem Assessing forecast quality for Expected Shortfall (ES) risk measure.
method Developed three ES encompassing test variants using joint loss functions.
result Tests robust to misspecification and effective in finite samples.
This paper describes an empirical study of shortfall optimization with Barra Extreme Risk. We compare minimum shortfall to minimum variance portfolios in the US, UK, and Japanese equity markets using Barra Style Factors (Value, Growth, Momentum, etc.). We show that minimizing shortfall generally improves performance ov…
The issue of constructing a risk minimizing hedge under an additional almost-surely type constraint on the shortfall profile is examined. Several classical risk minimizing problems are adapted to the new setting and solved. In particular, the bankruptcy threat of optimal strategies appearing in the classical risk minim…
In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Valu…
This paper assesses tail risk and systemic risk in cryptocurrencies using expectiles and MES.
problem Quantifying tail risk and systemic risk in cryptocurrencies.
method The study uses expectiles and Marginal Expected Shortfall (MES) to assess tail risk and systemic risk of cryptocurrencies.
result The expectile-based approach and MES provide a dynamic method to evaluate the impact of single assets on systemic risk.
Risk measures for multivariate financial positions are studied in a utility-based framework. Under a certain incomplete preference relation, shortfall and divergence risk measures are defined as the optimal values of specific set minimization problems. The dual relationship between these two classes of multivariate ris…
New axioms justify ES without NRC, linking it to mean-ES portfolio selection.
problem Economic axioms for portfolio risk assessment and mean-ES portfolio selection.
method Introducing concentration aversion as an alternative to NRC, establishing axiomatic foundations.
result Concentration aversion uniquely characterizes the family of ES and provides new formulas.
Digital currencies and cryptocurrencies have hesitantly started to penetrate the investors, and the next step will be the regulatory risk management framework. We examine the Value-at-Risk and Expected Shortfall properties for the major digital currencies, Bitcoin, Ethereum, Litecoin, and Ripple. The methodology used i…
We offer a simplified proof for Expected Shortfall's dual representation.
problem The dual representation of Expected Shortfall.
method Basic properties of quantile functions.
result New proof of Expected Shortfall's subadditivity.
The paper introduces a new method for forecasting financial risk using quantile-based modeling.
problem Forecasting Value-at-Risk (VaR) and Expected Shortfall (ES) for financial returns.
method Semiparametric approach using restricted quantile regression to model the conditional scale of financial returns.
result The method provides robust, distribution-free estimates of extreme losses and captures risk dynamics.