Undergraduate Honors Thesis Presentation: Using Expected Shortfall to Solve Risk Capital Allocation

This paper develops a coherent and statistically efficient framework for risk capital allocation by
replacing Value-at-Risk (VaR) with Expected Shortfall (ES) within the Constrained
Aumann–Shapley (CAS) allocation model. While VaR remains widely used in practice, its lack of
coherence and failure to properly account for tail risk can lead to unstable and economically
inconsistent capital allocations. In contrast, ES captures the full severity of tail losses and
satisfies the key axioms required for stable allocation.
           Building on this foundation, the paper characterizes component Expected Shortfall (CES) as a
pointwise conditional expectation, providing both economic interpretability and compatibility with
marginal allocation principles. However, estimating CES presents significant challenges due to
data sparsity in the tail region. To address this, we propose a class of nonparametric and
semi-parametric estimators, including kernel-based (Nadaraya–Watson) and local-linear
methods, enhanced with importance sampling via exponential tilting to efficiently target rare tail
events. 
           Using a controlled Gaussian simulation framework, we evaluate estimator performance in terms
of bias, variance, and RMSE. The results show that combining local-linear smoothing with
importance sampling substantially reduces estimation error, outperforming traditional
approaches that rely on naive Monte Carlo sampling. In particular, importance sampling
effectively increases tail sample efficiency, while local-linear methods mitigate boundary bias at
the ES threshold.
           Overall, this work provides both theoretical and computational contributions to modern risk
management by demonstrating that ES-based allocation not only restores coherence and
stability, but can also be implemented efficiently using advanced statistical techniques. These
findings are particularly relevant in light of regulatory frameworks such as the Fundamental
Review of the Trading Book (FRTB), which mandate ES as the standard risk measure

Thesis Advisor: José E. Figueroa-López