tail risk measurement

Gaussian models severely underestimate risk when heavy-tailed return distributions coexist with behavioral probability weighting biases. Method: This paper develops an econometric framework that jointly incorporates infinite divisibility and behavioral probability weighting. It innovatively couples a bounded probability weighting function with the Student’s *t* distribution—a heavy-tailed, infinitely divisible distribution—and proposes a joint estimation method for parameter inference. Contribution/Results: The framework simultaneously captures extreme asset return risks (via heavy tails) and nonlinear investor probability distortions (via behavioral weighting). Empirical analysis across 86 assets and over 430,000 daily observations shows that the model significantly outperforms the Gaussian benchmark in 88.4% of samples. At the 99% quantile, Value-at-Risk (VaR) underestimation declines sharply from 19.7% to 3.2%. Moreover, the estimator exhibits strong statistical properties, including consistency and asymptotic normality.

This paper addresses the comparative evaluation of financial tail-risk forecasting models—specifically Value-at-Risk (VaR) and Expected Shortfall (ES)—at extreme quantiles (e.g., 1%). It identifies critical failures of the Diebold–Mariano (DM) test and Hansen’s Model Confidence Set (MCS) under small samples (≤2 years) and highly asymmetric loss functions. Integrating Gneiting’s framework of strictly consistent scoring functions, the study employs Monte Carlo simulations and theoretical analysis to demonstrate that time-varying volatility and asymmetric loss jointly induce severe skewness in test statistics, distorting both standard and bootstrap critical values and drastically reducing statistical power—leading to frequent Type III errors. This work is the first to systematically characterize this dual-driver mechanism, rigorously delineates the validity boundaries of DM and MCS for tail-risk evaluation, and proposes diagnostic criteria to flag invalid applications alongside concrete directions for methodological improvement.

Accurately forecasting financial volatility and return quantiles (e.g., Value-at-Risk, Expected Shortfall) is critical for tail-risk assessment; however, conventional stochastic volatility (SV) models struggle to jointly capture skewness and heavy tails. To address this, we propose a novel class of Bayesian SV models that integrate realized volatility with three parameterizations of the skew-t distribution—including two newly introduced variants featuring skew-normal structures. This work is the first to combine realized volatility with a two-parameter skew-normal-type skew-t distribution, enabling simultaneous and flexible modeling of return skewness and kurtosis. Model estimation employs Markov Chain Monte Carlo (MCMC). Empirical analysis on U.S. and Japanese equity indices demonstrates that our models consistently outperform standard benchmarks in both volatility and quantile forecasting, significantly improving calibration and robustness of tail-risk measures.

Existing probabilistic forecasting evaluation methods lack the ability to characterize tail calibration—critical for high-impact extreme events, whose reliability is increasingly vital for risk-informed decision-making. Method: This paper introduces, for the first time, a general definition of tail calibration, rigorously connecting it to classical probabilistic calibration theory and integrating the Peaks-over-Threshold (POT) framework from extreme value theory. We develop an operational diagnostic framework by unifying probabilistic calibration theory, extreme-value statistics, diagnostic statistical tests, and empirical analysis. Contribution/Results: Applied to European precipitation forecasts, our framework significantly improves the quantification of predictive credibility for high-impact, rare events. It enables rigorous assessment of tail behavior in probabilistic forecasts and establishes a novel paradigm for extreme-event risk assessment and decision support.

This study addresses the joint identifiability and elicitability of tail risk measures—including Value-at-Risk (VaR), Expected Shortfall (ES), and Range Value-at-Risk (RVaR)—along with their associated quantiles. We establish, for the first time, necessary and sufficient conditions for their joint identifiability and elicitability. Methodologically, we construct a novel class of weighted scoring functions that uniformly generalizes the Fissler–Ziegel scoring family, enabling elicitation of previously non-elicitable functionals such as tail expectations conditional on quantiles. Our approach integrates distributional generators, generalized method of moments estimation, and regression modeling. The results provide a rigorous statistical foundation for tail risk modeling, substantially simplifying regression fitting, model comparison, and backtesting procedures. By ensuring coherent and robust evaluation of tail risk, this work enhances both the theoretical soundness and practical applicability of financial risk measurement.

This study addresses the bias inherent in tail index estimation for heavy-tailed distributions by proposing a novel estimator that integrates bias correction with empirical likelihood. The method uniquely combines bias correction techniques within an empirical likelihood framework to yield a more accurate and stable estimator, accompanied by rigorous asymptotic theory. Simulation experiments demonstrate that the proposed approach significantly outperforms existing methods in finite samples, while empirical analyses on real-world data further confirm its practical effectiveness and applicability.

This study addresses the challenge of discriminating and ranking predictive models based on their ability to capture tail behavior in heavy-tailed settings, such as insurance claim severity. The authors propose a novel tail scoring rule grounded in normalized upper order statistics, which, for the first time, integrates this statistic with proper scoring rules for tail inference. Theoretical analysis establishes the consistency and asymptotic normality of the proposed score and demonstrates that the classical Hill estimator arises as a special case. Simulations confirm the method’s effectiveness in distinguishing between distinct tail behaviors, while an application to real-world auto insurance claims data successfully enables comparative assessment and ranking of competing models according to their tail predictive performance.

This study investigates the nature of kurtosis in multivariate normal variance-mean mixture distributions, revealing that it arises from the interplay among mean variation, covariance structure, and the random mixing variable. By leveraging fourth-order cumulants, the paper presents the first decomposition of multivariate kurtosis into three interpretable components: directional, direction–covariance interaction, and covariance-pair terms, and establishes theoretical connections to Mardia’s kurtosis measure and projection pursuit. The proposed framework demonstrates that kurtosis is not merely a manifestation of tail behavior in specific directions but rather an aggregate signature of multiple sources of non-Gaussianity. Building on this insight, the authors develop novel diagnostic tools for assessing non-Gaussianity, identifying dominant tail directions, and detecting influential tail events, with empirical validation provided through simulations and daily stock return data.

Traditional normalizing flows struggle to capture the heavy-tailed nature of financial returns, leading to biased estimates of Value-at-Risk (VaR) and Expected Shortfall (ES). This work proposes Lévy-Flow, the first framework to integrate Lévy-driven heavy-tailed distributions—specifically Variance Gamma (VG) and Normal-Inverse Gaussian (NIG)—into normalizing flows. The model explicitly captures tail behavior while preserving exact likelihood computation and enabling efficient reparameterized sampling. Theoretically, it is shown that the proposed flow maintains the tail index under asymptotically linear transformations, which motivates the design of an Identity-tail Neural Spline Flow to faithfully preserve the base distribution’s tail shape. Empirical results on S&P 500 daily returns demonstrate that the VG flow reduces test negative log-likelihood by 69% compared to Gaussian flows and achieves well-calibrated 95% VaR, while the NIG flow yields the most accurate ES estimates.

This study addresses the challenges of modeling risks in Taiwan-exposed ETFs listed in the U.S.—notably heavy-tailed distributions, volatility clustering, and asymmetric responses to negative shocks—stemming from their concentration in technology stocks and exposure to geopolitical and supply chain disruptions. Integrating tail risk diagnostics via Hill estimation, asymmetric volatility modeling through GJR-GARCH, and portfolio optimization under both mean–variance and Conditional Value-at-Risk (CVaR) frameworks, the analysis of 30 ETFs reveals that differences in extreme downside risk arise primarily from scale parameters rather than tail indices. CVaR-based optimization yields substantially more concentrated allocations, and optimal portfolios vary markedly across performance metrics such as the Sharpe, STARR, and Rachev ratios. Empirical results indicate semiconductor ETFs exhibit significantly higher risk than diversified benchmarks, with the CVaR-efficient portfolio heavily overweighting SMH during the AI expansion phase, underscoring the limitations of traditional variance-based approaches in capturing risks of technology-intensive assets.