Monotone forward expected shortfall paths with joint conformal calibration

Çağlar Sözen; Çağlar Sözen

doi:10.3934/math.20261106

AIMS Mathematics

2026, Volume 11, Issue 9: 27665-27688. doi: 10.3934/math.20261106

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Monotone forward expected shortfall paths with joint conformal calibration

Çağlar Sözen ^,

Görele School of Applied Sciences, Department of Finance and Banking, Giresun University, Giresun, Turkey

Received: 06 November 2025 Revised: 25 February 2026 Accepted: 03 March 2026 Published: 01 September 2026
MSC : 62M10, 62P20, 91G70

We proposed a path-level framework for forecasting forward Expected Shortfall (ES) over multiple horizons using a leak-safe, shape-preserving, and conformal-calibrated design. A single multi-output predictor produced the full ES trajectory, and an isotonic post-projection enforced monotonicity in the reported multi-horizon ES path representation, ensuring horizon-wise coherence by construction under our target definition and sign convention. To quantify path-level uncertainty, we attached one-sided conformal safety bands using (ⅰ) a global scalar surcharge calibrated on the pathwise residual maximum and (ⅱ) a Šidák-adaptive allocation that redistributed the conformal error budget across horizons. Our empirical analysis focused on two representative exchange-traded funds (ETFs) with contrasting risk profiles, the iShares iBoxx $ \$ $ Investment Grade Corporate Bond ETF (LQD) and the ARK Innovation ETF (ARKK), under a strictly chronological label-end split with an $ H $-embargo and dependence-aware block calibration in the main specification ($ H = 20 $). The results showed that joint path coverage under the global safety band was near the nominal target for LQD and conservative for ARKK in the test sample, while the Šidák-adaptive safety band can materially reduce average excess buffer area (AEB; capital intensity) with only limited softening of joint path coverage in some cases. We also documented that marginal (horizon-wise) coverage can remain high even when joint path coverage differs, reinforcing the need to evaluate reliability at the path level. Appendix diagnostics further showed that calibration design choices (block vs. day-wise) can materially affect global-band joint path coverage, and that finite-sample Šidák and Bonferroni implementations may coincide under exact order-statistic quantiles when compared under matched nominal family-wise targets. Overall, the framework provided a transparent and modular way to move from single-horizon tail-risk summaries toward trajectory-level ES monitoring, while keeping assumptions and finite-sample limitations explicit.
- expected shortfall,
- multi-horizon forecasting,
- isotonic regression,
- conformal prediction,
- joint path coverage
Citation: Çağlar Sözen. Monotone forward expected shortfall paths with joint conformal calibration[J]. AIMS Mathematics, 2026, 11(9): 27665-27688. doi: 10.3934/math.20261106

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Abstract

We proposed a path-level framework for forecasting forward Expected Shortfall (ES) over multiple horizons using a leak-safe, shape-preserving, and conformal-calibrated design. A single multi-output predictor produced the full ES trajectory, and an isotonic post-projection enforced monotonicity in the reported multi-horizon ES path representation, ensuring horizon-wise coherence by construction under our target definition and sign convention. To quantify path-level uncertainty, we attached one-sided conformal safety bands using (ⅰ) a global scalar surcharge calibrated on the pathwise residual maximum and (ⅱ) a Šidák-adaptive allocation that redistributed the conformal error budget across horizons. Our empirical analysis focused on two representative exchange-traded funds (ETFs) with contrasting risk profiles, the iShares iBoxx $ \$ $ Investment Grade Corporate Bond ETF (LQD) and the ARK Innovation ETF (ARKK), under a strictly chronological label-end split with an $ H $-embargo and dependence-aware block calibration in the main specification ($ H = 20 $). The results showed that joint path coverage under the global safety band was near the nominal target for LQD and conservative for ARKK in the test sample, while the Šidák-adaptive safety band can materially reduce average excess buffer area (AEB; capital intensity) with only limited softening of joint path coverage in some cases. We also documented that marginal (horizon-wise) coverage can remain high even when joint path coverage differs, reinforcing the need to evaluate reliability at the path level. Appendix diagnostics further showed that calibration design choices (block vs. day-wise) can materially affect global-band joint path coverage, and that finite-sample Šidák and Bonferroni implementations may coincide under exact order-statistic quantiles when compared under matched nominal family-wise targets. Overall, the framework provided a transparent and modular way to move from single-horizon tail-risk summaries toward trajectory-level ES monitoring, while keeping assumptions and finite-sample limitations explicit.

References

[1]	A. J. McNeil, R. Frey, P. Embrechts, Quantitative risk management: Concepts, techniques and tools, Princeton University Press, 2015.
[2]	P. Embrechts, M. Hofert, Statistics and quantitative risk management for banking and insurance, Annu. Rev. Stat. Appl., 1 (2014), 493–514. https://doi.org/10.1146/annurev-statistics-022513-115631 doi: 10.1146/annurev-statistics-022513-115631
[3]	J. Danielsson, The illusion of control: Why financial crises happen, and what we can (and can't) do about it, Yale University Press, 2022.
[4]	M. Marcellino, J. H. Stock, M. W. Watson, A comparison of direct and iterated multistep AR methods for forecasting macroeconomic time series, J. Econ., 135 (2006), 499–526. https://doi.org/10.1016/j.jeconom.2005.07.020 doi: 10.1016/j.jeconom.2005.07.020
[5]	E. Ghysels, A. Sinko, R. Valkanov, MIDAS regressions: Further results and new directions, Econ. Rev., 26 (2007), 53–90. https://doi.org/10.1080/07474930600972467 doi: 10.1080/07474930600972467
[6]	E. Ghysels, M. Marcellino, Applied economic forecasting using time series methods, Oxford University Press, 2018.
[7]	J. Baruník, M. Nevrla, Quantile spectral beta: A tale of tail risks, investment horizons, and asset prices, J. Financ. Econ., 21 (2023), 1590–1646. https://doi.org/10.1093/jjfinec/nbac017 doi: 10.1093/jjfinec/nbac017
[8]	V. Vovk, A. Gammerman, G. Shafer, Algorithmic learning in a random world, Springer, 2005.
[9]	J. Lei, M. G'Sell, A. Rinaldo, R. J. Tibshirani, L. Wasserman, Distribution-free predictive inference for regression, J. Am. Stat. Assoc., 113 (2018), 1094–1111. https://doi.org/10.1080/01621459.2017.1307116 doi: 10.1080/01621459.2017.1307116
[10]	G. Shafer, V. Vovk, A tutorial on conformal prediction, J. Mach. Learn. Res., 9 (2008), 371–421.
[11]	A. N. Angelopoulos, S. Bates, Conformal prediction: A gentle introduction, Found. Trends Mach. Le., 16 (2023), 494–591. https://doi.org/10.1561/2200000101 doi: 10.1561/2200000101
[12]	R. F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation, Econometrica, 50 (1982), 987–1007. https://doi.org/10.2307/1912773 doi: 10.2307/1912773
[13]	T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, J. Econ., 31 (1986), 307–327. https://doi.org/10.1016/0304-4076(86)90063-1 doi: 10.1016/0304-4076(86)90063-1
[14]	T. G. Andersen, T. Bollerslev, F. X. Diebold, P. Labys, Modeling and forecasting realized volatility, Econometrica, 71 (2003), 579–625. https://doi.org/10.1111/1468-0262.00418 doi: 10.1111/1468-0262.00418
[15]	P. R. Hansen, Z. Huang, H. H. Shek, Realized GARCH: A joint model for returns and realized measures of volatility, J. Appl. Economet., 27 (2012), 877–906. https://doi.org/10.1002/jae.1234 doi: 10.1002/jae.1234
[16]	R. F. Engle, S. Manganelli, CAViaR: Conditional autoregressive value at risk by regression quantiles, J. Bus. Econ. Stat., 22 (2004), 367–381. https://doi.org/10.1198/073500104000000370 doi: 10.1198/073500104000000370
[17]	C. Acerbi, D. Tasche, Expected shortfall: A natural coherent alternative to value at risk, Econ. Notes, 31 (2002), 379–388. https://doi.org/10.1111/1468-0300.00091 doi: 10.1111/1468-0300.00091
[18]	T. Fissler, J. F. Ziegel, Higher order elicitability and Osband's principle, Ann. Statist., 44 (2016), 1680–1707. https://doi.org/10.1214/16-AOS1439 doi: 10.1214/16-AOS1439
[19]	F. X. Diebold, C. Li, Forecasting the term structure of government bond yields, J. Economet., 130 (2006), 337–364. https://doi.org/10.1016/j.jeconom.2005.03.005 doi: 10.1016/j.jeconom.2005.03.005
[20]	Y. Romano, E. Patterson, E. J. Candès, Conformalized quantile regression, Adv. Neural Inf. Process. Syst., 32 (2019), 3543–3553.
[21]	C. Xu, Y. Xie, Conformal prediction interval for dynamic time-series, In: Proceedings of the 38th International Conference on Machine Learning, 139 (2021), 11559–11569.
[22]	Ç. Sözen, F. Kabakcı, Forecasting future realized variance paths with depth-weighted ridge and conformal diagnostics, AIMS Mathematics, 10 (2025), 30246–30270. https://doi.org/10.3934/math.20251329 doi: 10.3934/math.20251329
[23]	Ç. Sözen, Uniform one-sided conformal bands for forward realized volatility curves, AIMS Mathematics, 10 (2025), 27314–27337. https://doi.org/10.3934/math.20251201 doi: 10.3934/math.20251201
[24]	Yahoo Finance, Historical market data for equities and ETFs (Adjusted Close series), 2025.
[25]	R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. Brunk, Statistical inference under order restrictions: The theory and application of isotonic regression, John Wiley & Sons, London, 1972.
[26]	T. Hastie, R. Tibshirani, J. Friedman, The elements of statistical learning: Data mining, inference, and prediction, Springer, 2009. https://doi.org/10.1007/978-0-387-84858-7
[27]	M. J. Best, N. Chakravarti, Active set algorithms for isotonic regression: A unifying framework, Math. Program., 47 (1990), 425–439. https://doi.org/10.1007/BF01580873 doi: 10.1007/BF01580873
[28]	T. Loughran, B. McDonald, When is a liability not a liability? Textual analysis, dictionaries, and 10-Ks, J. Financ., 66 (2011), 35–65. https://doi.org/10.1111/j.1540-6261.2010.01625.x doi: 10.1111/j.1540-6261.2010.01625.x
[29]	Z. Šidák, Rectangular confidence regions for the means of multivariate normal distributions, J. Am. Stat. Assoc., 62 (1967), 626–633. https://doi.org/10.1080/01621459.1967.10482935 doi: 10.1080/01621459.1967.10482935
[30]	S. Giannelos, I. Konstantelos, D. Pudjianto, G. Strbac, The impact of electrolyser allocation on Great Britain's electricity transmission system in 2050, Int. J. Hydrogen Energ., 202 (2026), 153097. https://doi.org/10.1016/j.ijhydene.2025.153097 doi: 10.1016/j.ijhydene.2025.153097
[31]	European Commission, E-mobility deployment and impact on grids: Impact of EV and charging infrastructure on European T & D grids: innovation needs, 2022. https://doi.org/10.2833/937755
[32]	Z. Dong, X. Zhang, L. Zhang, S. Giannelos, G. Strbac, Flexibility enhancement of urban energy systems through coordinated space heating aggregation of numerous buildings, Appl. Energ., 374 (2024), 123971. https://doi.org/10.1016/j.apenergy.2024.123971 doi: 10.1016/j.apenergy.2024.123971
[33]	S. Giannelos, I. Konstantelos, G. Strbac, Optimal supply chain design using machine learning, risk assessment and optimisation applied to coal distribution, EURO J. Decis. Processes, 13 (2025), 100062. https://doi.org/10.1016/j.ejdp.2025.100062 doi: 10.1016/j.ejdp.2025.100062

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