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.
New SigSwap model for path-dependent financial risk.
problem Managing complex, path-dependent financial risks.
method Geometry-based approach using path-signature and Signature Expected Shortfall.
result Path-dependent risks can be converted into transparent risk factors.
Expectile bears some interesting properties in comparison to the industry wide expected shortfall in terms of assessment of tail risk. We study the relationship between expectile and expected shortfall using duality results and the link to optimized certainty equivalent. Lower and upper bounds of expectile are derived …
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.
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.
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.
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.
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…
The problem of estimation error of Expected Shortfall is analyzed, with a view of its introduction as a global regulatory risk measure.
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.
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.
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.
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.
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. 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…
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.
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.
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.
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.
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…
We propose a new backtesting framework for Expected Shortfall that could be used by the regulator. Instead of looking at the estimated capital reserve and the realised cash-flow separately, one could bind them into the secured position, for which risk measurement is much easier. Using this simple concept combined with …
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.
The paper revisits expected signatures in semimartingale models, providing new formulae and simplifying complexity.
problem Computing expected signatures in semimartingale models.
method Revisits and provides new formulae for computing expected signatures in a general semimartingale setting.
result Log-transform of expected signatures simplifies complexity, leading to signature cumulants.
For a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the join…
The contour map of estimation error of Expected Shortfall (ES) is constructed. It allows one to quantitatively determine the sample size (the length of the time series) required by the optimization under ES of large institutional portfolios for a given size of the portfolio, at a given confidence level and a given esti…
A new method tests Expected Shortfall by analyzing both duration and severity of VaR violations.
problem Lack of separate testing for frequency and severity in ES backtesting.
method Uses bivariate orthogonal polynomials to derive moment conditions for durations and severities.
result Proposes a Wald test for identifying mis-specified components in ES models.
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.
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…
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…
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.
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.
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
We show that coherent risk measures are ineffective in curbing the behaviour of investors with limited liability or excessive tail-risk seeking behaviour if the market admits statistical arbitrage opportunities which we term ρ-arbitrage for a risk measure ρ. We show how to determine analytically whether such ρ-ar…
We introduce and compare new variability measures based on risk quantiles.
problem Comparing variability measures in risk management.
method Developed a framework for one-parameter families of inter-Expected Shortfall differences and inter-expectile differences.
result Characterized symmetric and comonotonic variability measures as mixtures of inter-Expected Shortfall differences.
We discuss the coherence properties of Expected Shortfall (ES) as a financial risk measure. This statistic arises in a natural way from the estimation of the "average of the 100p % worst losses" in a sample of returns to a portfolio. Here p is some fixed confidence level. We also compare several alternative representat…
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.
This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the only risk measures that satisfy a set of economic axioms for the Choquet expected …
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.
Optimal tontine strategy maximizes withdrawals while minimizing shortfall.
problem Maximizing withdrawals from a tontine account with withdrawal constraints.
method Dynamic programming and Fourier methods to solve PIDE, tested with historical data.
result Tontine overlay strategy outperforms constant withdrawal strategies.
Algorithm reduces historical expected shortfall computation by focusing on worst-case scenarios.
problem Computing the historical expected shortfall efficiently and accurately.
method Multi-step algorithm using Monte Carlo simulations to identify and reduce the number of worst-case scenarios.
result Non-asymptotic bounds for the L p-error of the expected shortfall estimator are derived.
Expected signatures map data streams to lower dimensions, improving ML performance.
problem Leveraging model-free embeddings for domain-agnostic machine learning.
method Expected signatures map data streams to lower dimensions, with convergence results bridging empirical and theoretical estimators.
result A modified expected signature estimator with lower mean squared error for martingale processes.
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.
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…
This research improves forecasting and testing of risk contributions using Expected Shortfall.
problem Improving risk allocation and testing methods for regulatory standards.
method Developed a comprehensive framework for backtesting and forecasting Expected Shortfall contributions.
result Proposed a novel semiparametric model for forecasting dynamic Expected Shortfall contributions.
The paper proposes efficient methods to learn VaR and ES using neural networks and Monte Carlo simulations.
problem Learning conditional VaR and ES in non-parametric setups with heavy-tailed financial losses.
method Two-step approach using Rademacher bounds, neural network quantile regression, and least-squares regression.
result Efficient learning schemes for multiple VaRs and ES are developed.
We study the properties of Expected Shortfall from the point of view of financial risk management. This measure --- which emerges as a natural remedy in some cases where Value at Risk (VaR) is not able to distinguish portfolios which bear different levels of risk --- is indeed shown to have much better properties than …
Kernel for Lévy rough paths derived from PDE system.
problem Computing similarity measures for Lévy rough paths.
method Developed a PDE system for the expected signature of inhomogeneous Lévy processes.
result Gaussian martingales' expected signature kernel satisfies a Goursat PDE.