Extreme Risk Measurement and the Selection of Optimal Investment Horizons

Empirical Evidence from the S&P 500

DOI

10.2991/978-94-6463-992-6_32How to use a DOI?

Keywords

Extreme Risk Measurement; Optimal Investment Horizon; S&P 500; Modified Historical VaR; Modified Monte Carlo VaR; Mean Modified VaR Ratio

Abstract

Financial crises, such as the recent 2020 COVID-19 shock or the 2008 Global Financial Crisis, revealed that market risk models cannot fully capture tail risks in stressed environments. In this paper, we fill this hiatus to by combining risk estimation and forward-looking allocation choices in a new framework implemented for the S&P 500 Index (January 2001 to December 2024). Methodologically, we simultaneously apply Historical VaR and Monte Carlo VaR with GARCH(1,1)–t dynamics and t-Copula dependence to capture heavy-tailed returns, sector interdependencies; Z-score normalization and horizon orthogonalization to normalize the scales, and a new statistic, the Mean Modified VaR Ratio (MMVR), is devised to benchmark the relative accuracy and time consistency of Monte Carlo estimates against the historical counterparty. We carry out a rolling-window analysis by 8 sectors (one missing because of the inadequate sample size) and varying investments horizons from 30 to about 6 years (Table 1 shows that in these sectors the most optimal investment horizons are concentrated close to 34–36 months (i.e. about 2.8–3.0 years) which indicates medium-term holdings ( i.e. more than one month and less than two years)) stabilizes the MMVR close to unity; optimal horizon for Industrials sector is slightly higher at approximately 48 months (≈4.0 years) which indicates its cyclical nature and slower convergence of MMVR to 1; optimal holding period is higher for Health Care sector with approximately 51 months (≈4.2 years) that, for longer horizons, predictability in sector-wide risk was maintained under sectoral dynamics in volatility, which suggests a longer forecast horizon would be optimal for maintaining predictable risk within sectors (except, for example, Utilities where predictability held only for very long horizons); while, in contrast, for Consumer Discretionary, 31 months (≈2.6 years) proved to be relatively shorter than could be sustained with short-horizon risk predictability. Combined, these results demonstrate that the MMVR paradigm is an effective means to discern horizon dependent stability regimes, and has practical implications to design sector-specific portfolio horizons.

Copyright

© 2026 The Author(s)

Open Access

Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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TY  - CONF
AU  - Jiashen Cui
PY  - 2026
DA  - 2026/02/20
TI  - Extreme Risk Measurement and the Selection of Optimal Investment Horizons
BT  - Proceedings of the 2025 4th International Conference on Mathematical Statistics and Economic Analysis (MSEA 2025)
PB  - Atlantis Press
SP  - 338
EP  - 360
SN  - 2352-5428
UR  - https://doi.org/10.2991/978-94-6463-992-6_32
DO  - 10.2991/978-94-6463-992-6_32
ID  - Cui2026
ER  -