Expected Shortfall

Forecasting the U.S. Treasury Yield Curve: A Distributionally Robust Machine Learning Approach ArXiv ID: 2601.04608 “View on arXiv” Authors: Jinjun Liu, Ming-Yen Cheng Abstract We study U.S. Treasury yield curve forecasting under distributional uncertainty and recast forecasting as an operations research and managerial decision problem. Rather than minimizing average forecast error, the forecaster selects a decision rule that minimizes worst case expected loss over an ambiguity set of forecast error distributions. To this end, we propose a distributionally robust ensemble forecasting framework that integrates parametric factor models with high dimensional nonparametric machine learning models through adaptive forecast combinations. The framework consists of three machine learning components. First, a rolling window Factor Augmented Dynamic Nelson Siegel model captures level, slope, and curvature dynamics using principal components extracted from economic indicators. Second, Random Forest models capture nonlinear interactions among macro financial drivers and lagged Treasury yields. Third, distributionally robust forecast combination schemes aggregate heterogeneous forecasts under moment uncertainty, penalizing downside tail risk via expected shortfall and stabilizing second moment estimation through ridge regularized covariance matrices. The severity of the worst case criterion is adjustable, allowing the forecaster to regulate the trade off between robustness and statistical efficiency. Using monthly data, we evaluate out of sample forecasts across maturities and horizons from one to twelve months ahead. Adaptive combinations deliver superior performance at short horizons, while Random Forest forecasts dominate at longer horizons. Extensions to global sovereign bond yields confirm the stability and generalizability of the proposed framework. ...

Extending the application of dynamic Bayesian networks in calculating market risk: Standard and stressed expected shortfall ArXiv ID: 2512.12334 “View on arXiv” Authors: Eden Gross, Ryan Kruger, Francois Toerien Abstract In the last five years, expected shortfall (ES) and stressed ES (SES) have become key required regulatory measures of market risk in the banking sector, especially following events such as the global financial crisis. Thus, finding ways to optimize their estimation is of great importance. We extend the application of dynamic Bayesian networks (DBNs) to the estimation of 10-day 97.5% ES and stressed ES, building on prior work applying DBNs to value at risk. Using the S&P 500 index as a proxy for the equities trading desk of a US bank, we compare the performance of three DBN structure-learning algorithms with several traditional market risk models, using either the normal or the skewed Student’s t return distributions. Backtesting shows that all models fail to produce statistically accurate ES and SES forecasts at the 2.5% level, reflecting the difficulty of modeling extreme tail behavior. For ES, the EGARCH(1,1) model (normal) produces the most accurate forecasts, while, for SES, the GARCH(1,1) model (normal) performs best. All distribution-dependent models deteriorate substantially when using the skewed Student’s t distribution. The DBNs perform comparably to the historical simulation model, but their contribution to tail prediction is limited by the small weight assigned to their one-day-ahead forecasts within the return distribution. Future research should examine weighting schemes that enhance the influence of forward-looking DBN forecasts on tail risk estimation. ...

Informative Risk Measures in the Banking Industry: A Proposal based on the Magnitude-Propensity Approach ArXiv ID: 2511.21556 “View on arXiv” Authors: Michele Bonollo, Martino Grasselli, Gianmarco Mori, Havva Nilsu Oz Abstract Despite decades of research in risk management, most of the literature has focused on scalar risk measures (like e.g. Value-at-Risk and Expected Shortfall). While such scalar measures provide compact and tractable summaries, they provide a poor informative value as they miss the intrinsic multivariate nature of risk.To contribute to a paradigmatic enhancement, and building on recent theoretical work by Faugeras and Pagés (2024), we propose a novel multivariate representation of risk that better reflects the structure of potential portfolio losses, while maintaining desirable properties of interpretability and analytical coherence. The proposed framework extends the classical frequency-severity approach and provides a more comprehensive characterization of extreme events. Several empirical applications based on real-world data demonstrate the feasibility, robustness and practical relevance of the methodology, suggesting its potential for both regulatory and managerial applications. ...

Levy-stable scaling of risk and performance functionals ArXiv ID: 2511.07834 “View on arXiv” Authors: Dmitrii Vlasiuk Abstract We develop a finite-horizon model in which liquid-asset returns exhibit Levy-stable scaling on a data-driven window [“tau_UV, tau_IR”] and aggregate into a finite-variance regime outside. The window and the tail index alpha are identified from the log-log slope of the central body and a two-segment fit of scale versus horizon. With an anchor horizon tau_0, we derive horizon-correct formulas for Value-at-Risk, Expected Shortfall, Sharpe and Information ratios, Kelly under a Value-at-Risk constraint, and one-step drawdown, where each admits a closed-form Gaussian-bias term driven by the exponent gap (1/alpha - 1/2). The implementation is nonparametric up to alpha and fixed tail quantiles. The formulas are reproducible across horizons on the Levy window. ...

Coherent estimation of risk measures ArXiv ID: 2510.05809 “View on arXiv” Authors: Martin Aichele, Igor Cialenco, Damian Jelito, Marcin Pitera Abstract We develop a statistical framework for risk estimation, inspired by the axiomatic theory of risk measures. Coherent risk estimators – functionals of P&L samples inheriting the economic properties of risk measures – are defined and characterized through robust representations linked to $L$-estimators. The framework provides a canonical methodology for constructing estimators with sound financial and statistical properties, unifying risk measure theory, principles for capital adequacy, and practical statistical challenges in market risk. A numerical study illustrates the approach, focusing on expected shortfall estimation under both i.i.d. and overlapping samples relevant for regulatory FRTB model applications. ...

Machine Learning Based Stress Testing Framework for Indian Financial Market Portfolios ArXiv ID: 2507.02011 “View on arXiv” Authors: Vidya Sagar G, Shifat Ali, Siddhartha P. Chakrabarty Abstract This paper presents a machine learning driven framework for sectoral stress testing in the Indian financial market, focusing on financial services, information technology, energy, consumer goods, and pharmaceuticals. Initially, we address the limitations observed in conventional stress testing through dimensionality reduction and latent factor modeling via Principal Component Analysis and Autoencoders. Building on this, we extend the methodology using Variational Autoencoders, which introduces a probabilistic structure to the latent space. This enables Monte Carlo-based scenario generation, allowing for more nuanced, distribution-aware simulation of stressed market conditions. The proposed framework captures complex non-linear dependencies and supports risk estimation through Value-at-Risk and Expected Shortfall. Together, these pipelines demonstrate the potential of Machine Learning approaches to improve the flexibility, robustness, and realism of financial stress testing. ...

Mirror Descent Algorithms for Risk Budgeting Portfolios ArXiv ID: 2411.12323 “View on arXiv” Authors: Unknown Abstract This paper introduces and examines numerical approximation schemes for computing risk budgeting portfolios associated to positive homogeneous and sub-additive risk measures. We employ Mirror Descent algorithms to determine the optimal risk budgeting weights in both deterministic and stochastic settings, establishing convergence along with an explicit non-asymptotic quantitative rate for the averaged algorithm. A comprehensive numerical analysis follows, illustrating our theoretical findings across various risk measures – including standard deviation, Expected Shortfall, deviation measures, and Variantiles – and comparing the performance with that of the standard stochastic gradient descent method recently proposed in the literature. ...

Some properties of Euler capital allocation ArXiv ID: 2405.00606 “View on arXiv” Authors: Unknown Abstract The paper discusses capital allocation using the Euler formula and focuses on the risk measures Value-at-Risk (VaR) and Expected shortfall (ES). Some new results connected to this capital allocation is known. Two examples illustrate that capital allocation with VaR is not monotonous which may be surprising since VaR is monotonous. A third example illustrates why the same risk measure should be used in capital allocation as in the evaluation of the total portfolio. We show how simulation may be used in order to estimate the expected Return on risk adjusted capital in the commitment period of an asset. Finally, we show how Markov chain Monte Carlo may be used in the estimation of the capital allocation. ...

Elicitability and identifiability of tail risk measures ArXiv ID: 2404.14136 “View on arXiv” Authors: Unknown Abstract Tail risk measures are fully determined by the distribution of the underlying loss beyond its quantile at a certain level, with Value-at-Risk, Expected Shortfall and Range Value-at-Risk being prime examples. They are induced by law-based risk measures, called their generators, evaluated on the tail distribution. This paper establishes joint identifiability and elicitability results of tail risk measures together with the corresponding quantile, provided that their generators are identifiable and elicitable, respectively. As an example, we establish the joint identifiability and elicitability of the tail expectile together with the quantile. The corresponding consistent scores constitute a novel class of weighted scores, nesting the known class of scores of Fissler and Ziegel for the Expected Shortfall together with the quantile. For statistical purposes, our results pave the way to easier model fitting for tail risk measures via regression and the generalized method of moments, but also model comparison and model validation in terms of established backtesting procedures. ...

A novel scaling approach for unbiased adjustment of risk estimators ArXiv ID: 2312.05655 “View on arXiv” Authors: Unknown Abstract The assessment of risk based on historical data faces many challenges, in particular due to the limited amount of available data, lack of stationarity, and heavy tails. While estimation on a short-term horizon for less extreme percentiles tends to be reasonably accurate, extending it to longer time horizons or extreme percentiles poses significant difficulties. The application of theoretical risk scaling laws to address this issue has been extensively explored in the literature. This paper presents a novel approach to scaling a given risk estimator, ensuring that the estimated capital reserve is robust and conservatively estimates the risk. We develop a simple statistical framework that allows efficient risk scaling and has a direct link to backtesting performance. Our method allows time scaling beyond the conventional square-root-of-time rule, enables risk transfers, such as those involved in economic capital allocation, and could be used for unbiased risk estimation in small sample settings. To demonstrate the effectiveness of our approach, we provide various examples related to the estimation of value-at-risk and expected shortfall together with a short empirical study analysing the impact of our method. ...