Mitigating optimistic bias in entropic risk estimation and optimization with an application to insurance (2409.19926v3)

Abstract: The entropic risk measure is widely used in high-stakes decision making to account for tail risks associated with an uncertain loss. With limited data, the empirical entropic risk estimator, i.e. replacing the expectation in the entropic risk measure with a sample average, underestimates the true risk. To mitigate the bias in the empirical entropic risk estimator, we propose a strongly asymptotically consistent bootstrapping procedure. The first step of the procedure involves fitting a distribution to the data, whereas the second step estimates the bias of the empirical entropic risk estimator using bootstrapping, and corrects for it. Two methods are proposed to fit a Gaussian Mixture Model to the data, a computationally intensive one that fits the distribution of empirical entropic risk, and a simpler one with a component that fits the tail of the empirical distribution. As an application of our approach, we study distributionally robust entropic risk minimization problems with type-$\infty$ Wasserstein ambiguity set and apply our bias correction to debias validation performance. Furthermore, we propose a distributionally robust optimization model for an insurance contract design problem that takes into account the correlations of losses across households. We show that choosing regularization parameters based on the 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. This improvement in performance can be explained from the observation that our methods suggest a higher (and more accurate) premium to homeowners.

• The paper presents a bootstrapping method that corrects the underestimation bias in empirical entropic risk measures using ERM matching and EVT techniques.
• It reformulates distributionally robust optimization with type-∞ Wasserstein ambiguity sets to achieve bounded worst-case risks via tractable convex models.
• Applied to insurance pricing, the approach improves premium calibration by incorporating rare tail events, significantly reducing out-of-sample risk.

Data-Driven Decision-Making Under Uncertainty with Entropic Risk Measure

The paper "Data-Driven Decision-Making Under Uncertainty with Entropic Risk Measure" by Utsav Sadana, Erick Delage, and Angelos Georghiou addresses the challenge of underestimating risk in high-stakes decision-making environments using the entropic risk measure (ERM). The authors proffer a novel bootstrapping technique to correct the biases observed in empirical estimations when limited data is available. Entropic risk estimation, distributionally robust optimization (DRO), and its application to insurance pricing problems are the pivotal areas of discussion.

Correcting Empirical Estimations of Entropic Risk

When using limited data, the empirical entropic risk estimator typically underestimates the true risk due to the omission of rare, but significant, tail events. The authors propose a strongly asymptotically consistent bootstrapping method to mitigate this bias. This method involves fitting a distribution to the available data followed by using this fitted distribution to perform bias-aware bootstrapping. Two distinct fitting methods are discussed:

1. Entropic Risk Matching (ERM): This method fits a Gaussian Mixture Model (GMM) to mimic the bias present in scenarios drawn from the empirical distribution. The authors employ Wasserstein distance to tune the GMM parameters, ensuring that the entropic risks derived from bootstrapped samples match those of empirical data.
2. Extreme Value Theory (EVT): The EVT method fits a two-component GMM by characterizing the distribution of maxima observed in the data blocks. This distribution aims to capture tail events accurately, which significantly influence the entropic risk measure.

Both methods demonstrate strong asymptotic consistency for bias correction, potentially making them versatile tools for broader classes of risk measures beyond ERM.

Distributionally Robust Optimization (DRO) Formulations

Traditional type-$p$ Wasserstein ambiguity sets ($p<\infty$) are shown to result in unbounded solutions when combined with the entropic risk criterion due to the non-linear growth in risk with distribution perturbations. Instead, the authors propose using the type-$\infty$ Wasserstein ambiguity sets, which lead to a bounded worst-case risk for robust decision-making. Specifically, they reformulate the DRO problem to a tractable convex optimization problem for piecewise concave loss functions.

The type-$\infty$ ambiguity set encapsulates the true essence of distributional robustness by considering worst-case deviations over the entire support of the empirical distribution, rather than being confined to finite perturbations. This leads to significant improvements in out-of-sample performance when applied to decision-making contexts like portfolio optimization or risk-averse utility maximization.

Application to Insurance Pricing

The proposed methods find an applied context in insurance pricing, particularly for designing catastrophe insurance policies. Traditional empirical estimations often lead to underpricing due to ignored tail risks. By using the authors' proposed bias-corrected CV approach, insurers can better calibrate premiums reflecting the true risk exposure.

Insurers face the challenge of pricing insurance contracts for multiple policyholders with correlated risks comprehensively. The DRO model equipped with type-$\infty$ Wasserstein sets allows for the incorporation of rare extreme events, ensuring that cross-validation of premiums and entropic risk measures yield significantly lower out-of-sample risks.

Numerical Experiments

The empirical results validate the theoretical contributions. Numerical experiments highlight that the proposed bias correction methods (BS-EVT and BS-Match) notably reduce the underestimation bias in entropic risk measures compared to traditional cross-validation and SAA approaches. In scenarios involving limited data and complex correlation structures between policyholders' risks, the proposed methods offer substantial improvements in predictive performance and generalizability.

Implications and Future Directions

The findings put forth in this paper have strong implications for both theoretical risk management and practical applications in financial risk, insurance, and operations research. By providing robust statistical tools to correct bias in risk estimation and optimization frameworks adaptable for high-dimensional, complex decision-making problems, this paper bridges a significant gap in the current literature.

Future work may involve extending these methods to other risk measures and decision-making frameworks, examining their applicability in real-time financial markets, and exploring computational enhancements to the fit-and-bootstrap procedure for improved scalability.

Overall, the contributions of this paper offer profound insights and practical tools, reinforcing the significance of robust, data-driven decision-making under uncertainty.