Simulating Confidence Intervals for Conditional Value-at-Risk via Least-Squares Metamodels

Metamodeling techniques have been applied to approximate portfolio loss as a function of financial risk factors, thus producing point estimates of various measures of portfolio risk based on Monte Carlo samples. Rather than point estimates, this paper focuses on the construction of confidence intervals (CIs) for a widely used risk measure, the so-called conditional value-at-risk (CVaR), when the least-squares method (LSM) is employed as a metamodel in the point estimation. To do so, we first develop lower and upper bounds of CVaR and construct CIs for these bounds. Then, the lower end of the CI for the lower bound and the upper end of the CI for the upper bound together form a CI of CVaR with justifiable statistical guarantee, which accounts for both the metamodel error and the noises of Monte Carlo samples. The proposed CI procedure reuses the samples simulated for LSM point estimation, thus requiring no additional simulation budget. We demonstrate via numerical examples that the proposed procedure may lead to a CI with the desired coverage probability and a much smaller width than that of an existing CI in the literature. History: Accepted by Bruno Tuffin, Area Editor for Simulation. Funding: This research was supported by the National Natural Science Foundation of China (NNSFC) [Grants 72101260 and 72471232], the Research Grants Council of Hong Kong (RGC-HK) [General Research Fund Project 11508620], InnoHK Initiative, the Government of the HKSAR, and Laboratory for AI-Powered Financial Technologies, and NNSFC/RGC-HK Joint Research Scheme [Project N_CityU 105/21]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0394 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0394 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .