Efficient and Adaptive Estimation of Portfolio Weights with Spectral Risk Measures
Abstract
Spectral risk measures, including conditional value-at-risk (CVaR), generate a family of convex criteria for portfolio estimation. We study how the criterion should be chosen when different criteria share the same population minimizer, and whether the efficient criterion can itself be learned from data. We develop a general asymptotic theory for empirical spectral-risk minimization under estimated linear constraints. Under normal scale-mixture elliptical returns, all spectral risk measures identify the same population efficient portfolio, but their empirical minimizers have different sampling distributions. Their asymptotic covariance decomposes into a common component and a positive-semidefinite component scaled by a functional of the spectral measure, reducing efficiency to an optimization over probability measures. We characterize the efficiency-optimal spectral measure and show that single-level CVaR is generally inefficient. We then construct a fully data-adaptive estimator that learns the radial distribution and optimal spectral measure from the same observations used for portfolio estimation, yet has the same first-order distribution as the infeasible oracle. Simulations and an empirical application illustrate the method.
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BibTeX
@misc{zhou2026efficient,
title = {{Efficient and Adaptive Estimation of Portfolio Weights with Spectral Risk Measures}},
author = {Kangjie Zhou and Ming Yuan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31994},
eprint = {2609.31994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31994},
}
RIS
TY - UNPB
TI - Efficient and Adaptive Estimation of Portfolio Weights with Spectral Risk Measures
AU - Kangjie Zhou
AU - Ming Yuan
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31994
UR - https://arxiv.org/abs/2609.31994
AB - Spectral risk measures, including conditional value-at-risk (CVaR), generate a family of convex criteria for portfolio estimation. We study how the criterion should be chosen when different criteria share the same population minimizer, and whether the efficient criterion can itself be learned from data. We develop a general asymptotic theory for empirical spectral-risk minimization under estimated linear constraints. Under normal scale-mixture elliptical returns, all spectral risk measures identify the same population efficient portfolio, but their empirical minimizers have different sampling distributions. Their asymptotic covariance decomposes into a common component and a positive-semidefinite component scaled by a functional of the spectral measure, reducing efficiency to an optimization over probability measures. We characterize the efficiency-optimal spectral measure and show that single-level CVaR is generally inefficient. We then construct a fully data-adaptive estimator that learns the radial distribution and optimal spectral measure from the same observations used for portfolio estimation, yet has the same first-order distribution as the infeasible oracle. Simulations and an empirical application illustrate the method.
ER -