Core Concepts

The empirical entropic risk estimator significantly underestimates the true risk, particularly for risk-averse decision makers. This paper proposes a strongly asymptotically consistent bootstrapping procedure to debias the empirical entropic risk estimator by fitting bias-aware distributions to the data.

Abstract

The paper addresses the challenge of accurately estimating entropic risk in high-stakes decision-making, where rare events and extreme losses are a significant concern. The entropic risk measure is widely used in applications such as finance, portfolio selection, and insurance pricing, but the empirical entropic risk estimator tends to underestimate the true risk, especially for risk-averse decision makers.

The authors propose a bias correction procedure that involves two steps:

- Fitting a distribution (e.g., Gaussian Mixture Model) to the data to capture the bias in the samples.
- Using bootstrapping to estimate the bias of the empirical entropic risk estimator and correcting for it.

The authors show that naively fitting a Gaussian Mixture Model using maximum likelihood estimation does not fully resolve the underestimation issue. To address this, they introduce two alternative methods:

- Entropic Risk Matching: Fitting the distribution of entropic risk directly, using a Wasserstein distance minimization approach.
- Extreme Value Theory (EVT) Matching: Approximating the distribution of the maxima of the data using a two-component Gaussian Mixture Model.

The authors also study a distributionally robust entropic risk minimization problem with a type-∞ Wasserstein ambiguity set, where debiasing the validation performance using their techniques significantly improves the calibration of the size of the ambiguity set.

Furthermore, the authors propose a distributionally robust optimization model for a well-studied insurance contract design problem, where they show that cross-validation methods can result in significantly higher out-of-sample risk for the insurer if the bias in validation performance is not corrected for.

To Another Language

from source content

arxiv.org

Stats

The true entropic risk is computed as: ρP(ξ) = 1
3 log
P5
y=1 πy exp
3µy + 9
2σ2
y
.
The empirical entropic risk is computed as: ρˆPN(ℓ(ξ)) = 1
α log( 1
N
PN
i=1 exp(αℓ(ˆξi))).

Quotes

"The entropic risk measure is useful in high-stakes decision-making, where rare events and their associated extreme losses are a significant concern."
"Even with a large number of samples, the empirical entropic risk may still significantly underestimate the true risk, particularly for risk-averse decision makers."

Key Insights Distilled From

by Utsav Sadana... at **arxiv.org** 10-01-2024

Deeper Inquiries

The proposed bias correction methods, particularly the strongly asymptotically consistent bootstrapping procedure, can be extended to other risk measures by adapting the underlying principles of bias estimation and correction. For instance, similar to the entropic risk measure, many risk measures, such as Value at Risk (VaR) and Conditional Value at Risk (CVaR), also suffer from underestimation issues when empirical distributions are used.
To extend the bias correction methods, one could follow these steps:
Identify the Risk Measure: Determine the specific risk measure to be analyzed, such as VaR or CVaR, and understand its mathematical formulation and properties.
Fit a Distribution: Just as the empirical entropic risk estimator uses a fitted distribution to estimate bias, the same approach can be applied to other risk measures. For example, fitting a Generalized Pareto Distribution (GPD) for tail risk estimation in the case of VaR.
Bootstrap from the Fitted Distribution: Implement a bootstrapping procedure that samples from the fitted distribution rather than the empirical distribution. This allows for a more accurate representation of tail risks, which are critical in risk measures like VaR and CVaR.
Estimate and Correct Bias: Calculate the bias associated with the empirical risk estimator and apply a correction term derived from the fitted distribution. This correction can be based on the observed discrepancies between the empirical and true risk measures, similar to the approach used for entropic risk.
By following these steps, the bias correction methods can be effectively adapted to enhance the accuracy of various risk measures, thereby improving decision-making processes across different domains.

The underestimation of entropic risk has significant implications across various applications, including portfolio optimization and revenue management.
Portfolio Optimization: In portfolio management, underestimating entropic risk can lead to suboptimal asset allocation decisions. Investors may take on excessive risk, believing their portfolios are less risky than they truly are. This misjudgment can result in substantial losses during market downturns, as the true tail risks associated with extreme market movements are not adequately accounted for. Consequently, this can undermine the long-term sustainability of investment strategies and lead to significant financial repercussions.
Revenue Management: In revenue management, businesses often rely on risk measures to set prices and manage inventory. If entropic risk is underestimated, companies may set prices too low, failing to capture the full potential of high-demand scenarios. This can lead to lost revenue opportunities, especially in industries with volatile demand patterns, such as airlines and hospitality. Additionally, underestimating risk can result in inadequate inventory levels, leading to stockouts or overstock situations that negatively impact profitability.
Overall Decision-Making: The broader implication of underestimating entropic risk is the potential for poor decision-making under uncertainty. In any domain where risk assessment is critical, such as finance, healthcare, and supply chain management, failing to accurately estimate risks can lead to misguided strategies, increased vulnerability to adverse events, and ultimately, a failure to achieve organizational objectives.

The insights from the proposed bias correction methods for entropic risk can be leveraged to enhance decision-making frameworks under uncertainty in several ways:
Incorporating Bias Correction in Risk Assessment: By adopting the bias correction techniques outlined in the study, decision-makers in various fields can improve the accuracy of their risk assessments. This involves fitting appropriate distributions to historical data and applying bootstrapping methods to estimate and correct biases in risk measures. Such practices can lead to more reliable risk evaluations, which are crucial for informed decision-making.
Developing Robust Optimization Models: The framework for distributionally robust optimization (DRO) introduced in the context of entropic risk can be applied to other domains, such as supply chain management and finance. By considering ambiguity in the underlying distributions of uncertain parameters, organizations can develop optimization models that are less sensitive to estimation errors and better equipped to handle worst-case scenarios.
Enhancing Predictive Analytics: The methodologies for estimating and correcting biases can be integrated into predictive analytics frameworks. By ensuring that predictive models account for tail risks and potential underestimations, organizations can improve their forecasting accuracy and make more informed strategic decisions.
Cross-Disciplinary Applications: The principles of bias correction and robust optimization can be adapted to various fields, including healthcare (for patient risk assessments), environmental management (for assessing risks of natural disasters), and public policy (for evaluating the impacts of policy decisions). This adaptability underscores the versatility of the proposed methods and their potential to enhance decision-making across diverse sectors.
By implementing these insights, organizations can create more robust and reliable decision-making frameworks that effectively navigate uncertainty, ultimately leading to better outcomes and enhanced resilience against adverse events.

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