Expected shortfall estimation and capital coverage under overlapping scenarios
Authors: Martin Aichele, Igor Cialenco, Damian Jelito, Marcin Pitera
Abstract
Expected shortfall (ES) is routinely estimated from a short sample of profit-and-loss (P&L) scenarios built from overlapping multi-day returns, while the non-parametric estimators applied to such data are designed under independent sampling. We study how this mismatch affects estimation accuracy and capital coverage, and how it interacts with serial dependence and heavy tails in the underlying P&L. Working within the coherent risk estimation framework, we compare six order-statistic-based ES estimators on fit, bias, and capital-coverage metrics. Following the Fundamental Review of the Trading Book (FRTB), we use a one-year estimation window and ES at the $2.5\%$ confidence level as a benchmark setup. We combine simulations with market data on twelve risk factors and ninety-nine equity portfolios. We show that overlapping sampling does not merely add noise but shifts ES estimates systematically downward, thereby weakening capital coverage. Serial dependence acts in the same direction, and heavy tails amplify both effects. More conservative weighting mitigates the underestimation but does not remove it. On market data, the confidence level at which the estimated capital covers realized losses roughly doubles in median under overlapping sampling for every estimator, while the $\textrm{ES}_{2.5\%}$-to-$\textrm{VaR}_{1\%}$ ratio, which links the FRTB base market risk capital metric to its pre-FRTB predecessor, changes comparatively little. Overlapping sampling can thus substantially understate ES-based capital. Since many banks apply a small set of standard estimators in their FRTB regulatory or economic capital model in a historical simulation approach, this understatement is likely to be common for many institutions, and combining simulation-based assessment of estimation risk with coverage-based diagnostics offers a practical method to detect it in model validation and monitoring.
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Risk measures induced by efficient insurance contracts characterizes a mixture of mean and expected shortfall for insurance, rather than estimating ES from overlapping scenarios.
Abstract
The Expected Shortfall (ES) is one of the most important regulatory risk measures in finance, insurance, and statistics, which has recently been characterized via sets of axioms from perspectives of portfolio risk management and statistics. Meanwhile, there is large literature on insurance design with ES as an objective or a constraint. A visible gap is to justify the special role of ES in insurance and actuarial science. To fill this gap, we study characterization of risk measures induced by efficient insurance contracts, i.e., those that are Pareto optimal for the insured and the insurer. One of our major results is that we characterize a mixture of the mean and ES as the risk measure of t…
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Forecast Evaluation in Large Cross-Sections of Realized VolatilityChristis Katsouris · 2021 · Machine Learning · Not summarized yet
Forecast Evaluation in Large Cross-Sections of Realized Volatility evaluates volatility forecasts under cross-sectional dependence, not ES estimation or capital coverage.
Abstract
In this paper, we consider the forecast evaluation of realized volatility measures under cross-section dependence using equal predictive accuracy testing procedures. We evaluate the predictive accuracy of the model based on the augmented cross-section when forecasting Realized Volatility. Under the null hypothesis of equal predictive accuracy the benchmark model employed is a standard HAR model while under the alternative of non-equal predictive accuracy the forecast model is an augmented HAR model estimated via the LASSO shrinkage. We study the sensitivity of forecasts to the model specification by incorporating a measurement error correction as well as cross-sectional jump component measur…
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A semi-parametric dynamic conditional correlation framework for risk forecastingGiuseppe Storti, Chao Wang · 2022 · Risk Management · cited 3 times · AI summary available
A semi-parametric dynamic conditional correlation framework forecasts portfolio VaR and ES by modeling asset dependence, unlike the paper’s comparison of nonparametric estimators.
Abstract
We develop a novel multivariate semi-parametric framework for joint portfolio Value-at-Risk (VaR) and Expected Shortfall (ES) forecasting. Unlike existing univariate semi-parametric approaches, the proposed framework explicitly models the dependence structure among portfolio asset returns through a dynamic conditional correlation (DCC) parameterization. To estimate the model, a two-step procedure based on the minimization of a strictly consistent VaR and ES joint loss function is employed. This procedure allows to simultaneously estimate the DCC parameters and the portfolio risk factors. The performance of the proposed model in risk forecasting on various probability levels is evaluated by m…
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Nonparametric Tests of Tail Behavior in Stochastic Frontier ModelsWilliam, C. Horrace, Yulong Wang · 2020 · Econometrics · cited 5 times · AI summary available
Nonparametric Tests of Tail Behavior in Stochastic Frontier Models tests tail properties of frontier-model errors, rather than ES estimation from financial scenarios.
Abstract
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Double Descent in Portfolio Optimization: Dance between Theoretical Sharpe Ratio and Estimation AccuracyYonghe Lu, Yanrong Yang, Terry Zhang · 2024 · Portfolio Management · cited 2 times · AI summary available
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Abstract
We study the relationship between model complexity and out-of-sample performance in the context of mean-variance portfolio optimization. Representing model complexity by the number of assets, we find that the performance of low-dimensional models initially improves with complexity but then declines due to overfitting. As model complexity becomes sufficiently high, the performance improves with complexity again, resulting in a double ascent Sharpe ratio curve similar to the double descent phenomenon observed in artificial intelligence. The underlying mechanisms involve an intricate interaction between the theoretical Sharpe ratio and estimation accuracy. In high-dimensional models, the theore…
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Risk sharing for capital requirements with multidimensional security marketsFelix-Benedikt Liebrich, Gregor Svindland · 2018 · Risk Management · cited 27 times · AI summary available
Risk sharing for capital requirements studies allocation and equilibrium under capital adequacy constraints, rather than estimation of ES-based capital.
Abstract
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Sample Efficient Deep Reinforcement Learning via Uncertainty EstimationVincent Mai, Kaustubh Mani, Liam Paull · 2022 · Machine Learning · cited 62 times · AI summary available
Sample Efficient Deep Reinforcement Learning via Uncertainty Estimation uses uncertainty-weighted learning to improve sample efficiency, outside financial risk estimation.
Abstract
In model-free deep reinforcement learning (RL) algorithms, using noisy value estimates to supervise policy evaluation and optimization is detrimental to the sample efficiency. As this noise is heteroscedastic, its effects can be mitigated using uncertainty-based weights in the optimization process. Previous methods rely on sampled ensembles, which do not capture all aspects of uncertainty. We provide a systematic analysis of the sources of uncertainty in the noisy supervision that occurs in RL, and introduce inverse-variance RL, a Bayesian framework which combines probabilistic ensembles and Batch Inverse Variance weighting. We propose a method whereby two complementary uncertainty estimatio…
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Portfolio liquidation under transient price impact -- theoretical solution and implementation with 100 NASDAQ stocksYing Chen, Ulrich Horst, Hoang Hai Tran · 2019 · Mathematical Finance · cited 12 times · AI summary available
Portfolio liquidation under transient price impact derives and tests trading strategies using NASDAQ data, rather than estimating tail risk or capital coverage.
Abstract
We derive an explicit solution for deterministic market impact parameters in the Graewe and Horst (2017) portfolio liquidation model. The model allows to combine various forms of market impact, namely instantaneous, permanent and temporary. We show that the solutions to the two benchmark models of Almgren and Chris (2001) and of Obizhaeva and Wang (2013) are obtained as special cases. We relate the different forms of market impact to the microstructure of limit order book markets and show how the impact parameters can be estimated from public market data. We investigate the numerical performance of the derived optimal trading strategy based on high frequency limit order books of 100 NASDAQ s…
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