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29 pages, 577 KiB

Open AccessArticle

Symmetric Adjustable Tail-Risk Measure for Distributionally Robust Optimization in Portfolio Allocation

by Haonan Wang, Yunxiao Zhao, Yixin Guo, Changhe Liu and Xinlin Zhang

Symmetry 2025, 17(6), 959; https://doi.org/10.3390/sym17060959 - 17 Jun 2025

Viewed by 397

In this study, we begin by extending the mathematical formulation of the expectile risk measure through a key modification: replacing the expectation in its defining equation with expected shortfall. This substitution leads to a revised risk measure that more precisely captures downside risk. [...] Read more.

In this study, we begin by extending the mathematical formulation of the expectile risk measure through a key modification: replacing the expectation in its defining equation with expected shortfall. This substitution leads to a revised risk measure that more precisely captures downside risk. To handle the uncertainty of the underlying distribution, we then adopt a distributionally robust optimization framework. Notably, this robust optimization problem can be reformulated as a linear programming problem, and by employing suitable approximation techniques, we derive an analytical solution. In numerical experiments, our portfolio problem exhibits superior performance when compared to several traditional and distributionally robust optimized portfolio problems. Full article

(This article belongs to the Special Issue Symmetry in Optimal Control and Applications)

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15 pages, 742 KiB

Open AccessFeature PaperArticle

Risk Measure Examination for Large Losses

by Miwaka Yamashita

Mathematics 2025, 13(12), 1974; https://doi.org/10.3390/math13121974 - 15 Jun 2025

Viewed by 328

Abstract

The risk measures such as value at risk, and conditional values at risk do not always account for the sensitivity of large losses with certainty, as large losses often break the homogeneity especially seen in an illiquidity risk. In this study, we examine [...] Read more.

The risk measures such as value at risk, and conditional values at risk do not always account for the sensitivity of large losses with certainty, as large losses often break the homogeneity especially seen in an illiquidity risk. In this study, we examine the characteristics of large-loss sensitivity more holistically, including small probability, within the framework of risk measures. The analysis incorporates the certainty equivalent, generation of the optimal certainty equivalent formulation, divergence utility, and general utility functions in their original form, and their relationship with expectiles and elicitability. The discussion provides a summary in the understanding of risk measure status and sensitivity involving small probably cases. Additionally, we evaluate large-loss sensitivity in risk-sharing scenarios using the convex conjugation of the divergence utility. By clarifying the conditions affecting large-loss sensitivity, the findings highlight the limitations of existing risk measures and suggest directions for future improvement. Furthermore, these insights contribute to enhancing the stability of risk-sharing business models. Full article

(This article belongs to the Special Issue Advances in Financial Mathematics and Risk Management)

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20 pages, 2808 KiB

Open AccessArticle

Nonparametric Estimation of Dynamic Value-at-Risk: Multifunctional GARCH Model Case

by Zouaoui Chikr-Elmezouar, Ali Laksaci, Ibrahim M. Almanjahie and Fatimah Alshahrani

Mathematics 2025, 13(12), 1961; https://doi.org/10.3390/math13121961 - 13 Jun 2025

Viewed by 376

Abstract

Value-at-Risk (VaR) estimation using the GARCH model is an important topic in financial data analysis. It allows for an increase in the accuracy of risk assessment by controlling time-varying volatility. In this paper, we enhance this feature by exploring the functional path of [...] Read more.

Value-at-Risk (VaR) estimation using the GARCH model is an important topic in financial data analysis. It allows for an increase in the accuracy of risk assessment by controlling time-varying volatility. In this paper, we enhance this feature by exploring the functional path of the financial data. More precisely, we study the nonparametric estimation of the multi-functional VaR function using the local linear method, construct an estimator, and establish its stochastic consistency. The derived asymptotic result provides a rigorous mathematical foundation that permits boosting the use of the VaR function in financial data analysis. Furthermore, an empirical analysis is performed in order to examine the efficiency of the proposed algorithm. Additionally, a real data application is created to highlight the multi-functionality of the VaR estimation for multi-asset risk management. Full article

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19 pages, 324 KiB

Open AccessArticle

Sparse Robust Weighted Expectile Screening for Ultra-High-Dimensional Data

by Xianjun Wu, Pingping Han and Mingqiu Wang

Axioms 2025, 14(5), 340; https://doi.org/10.3390/axioms14050340 - 28 Apr 2025

Viewed by 306

Abstract

This paper investigates robust feature screening for ultra-high dimensional data in the presence of outliers and heterogeneity. Considering the susceptibility of likelihood methods to outliers, we propose a Sparse Robust Weighted Expectile Regression (SRoWER) method that combines the

L_{2} E

criterion with [...] Read more.

This paper investigates robust feature screening for ultra-high dimensional data in the presence of outliers and heterogeneity. Considering the susceptibility of likelihood methods to outliers, we propose a Sparse Robust Weighted Expectile Regression (SRoWER) method that combines the

L_{2} E

criterion with expectile regression. By utilizing the IHT algorithm, our method effectively incorporates correlations of covariates and enables joint feature screening. The proposed approach demonstrates robustness against heavy-tailed errors and outliers in data. Simulation studies and a real data analysis are provided to demonstrate the superior performance of the SRoWER method when dealing with outlier-contaminated explanatory variables and/or heavy-tailed error distributions. Full article

(This article belongs to the Section Mathematical Physics)

18 pages, 1382 KiB

Open AccessArticle

Finite Mixture at Quantiles and Expectiles

by Marilena Furno

J. Risk Financial Manag. 2025, 18(4), 177; https://doi.org/10.3390/jrfm18040177 - 27 Mar 2025

Viewed by 268

Abstract

Finite mixture regression identifies homogeneous groups within a sample and computes the regression coefficients in each group. Groups and group coefficients are jointly estimated using an iterative approach. This work extends the finite mixture estimator to the tails of the distribution, by incorporating [...] Read more.

Finite mixture regression identifies homogeneous groups within a sample and computes the regression coefficients in each group. Groups and group coefficients are jointly estimated using an iterative approach. This work extends the finite mixture estimator to the tails of the distribution, by incorporating quantiles and expectiles and relaxing the constraint of constant group probability adopted in previous analysis. The probability of each group depends on the selected location: an observation can be allocated in the best-performing group if we look at low values of the dependent variable, while at higher values it may be assigned to the poorly performing class. We explore two case studies: school data from a PISA math proficiency test and asset returns from the Center for Research in Security Prices. In these real data examples, group classifications change based on the selected location of the dependent variable, and this has an impact on the regression estimates due to the joint computation of class probabilities and class regressions coefficients. A Monte Carlo experiment is conducted to compare the performances of the discussed estimators with results of previous research. Full article

(This article belongs to the Special Issue Machine Learning-Based Risk Management in Finance and Insurance)

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19 pages, 798 KiB

Open AccessArticle

Multifunctional Expectile Regression Estimation in Volterra Time Series: Application to Financial Risk Management

by Somayah Hussain Alkhaldi, Fatimah Alshahrani, Mohammed Kbiri Alaoui, Ali Laksaci and Mustapha Rachdi

Axioms 2025, 14(2), 147; https://doi.org/10.3390/axioms14020147 - 19 Feb 2025

Cited by 1 | Viewed by 779

Abstract

We aim to analyze the dynamics of multiple financial assets with variable volatility. Instead of a standard analysis based on the Black–Scholes model, we proceed with the multidimensional Volterra model, which allows us to treat volatility as a stochastic process. Taking advantage of [...] Read more.

We aim to analyze the dynamics of multiple financial assets with variable volatility. Instead of a standard analysis based on the Black–Scholes model, we proceed with the multidimensional Volterra model, which allows us to treat volatility as a stochastic process. Taking advantage of the long memory function of this type of model, we analyze the reproduced movements using recent algorithms in the field of functional data analysis (FDA). In fact, we develop, in particular, new risk tools based on the asymmetric least squares loss function. We build an estimator using the multifunctional kernel (MK) method and then establish its asymptotic properties. The multidimensionality of the Volterra process is explored through the dispersion component of the convergence rate, while the nonparametric path of the risk tool affects the bias component. An empirical analysis is conducted to demonstrate the ease of implementation of our proposed approach. Additionally, an application on real data is presented to compare the effectiveness of expectile-based measures with Value at Risk (VaR) in financial risk management for multiple assets. Full article

(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)

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27 pages, 376 KiB

Open AccessArticle

Improved Confidence Intervals for Expectiles

by Spiridon Penev and Yoshihiko Maesono

Mathematics 2025, 13(3), 510; https://doi.org/10.3390/math13030510 - 4 Feb 2025

Viewed by 679

Abstract

Expectiles were introduced to statistics around 40 years ago, but have recently gained renewed interest due to their relevance in financial risk management. In particular, the 2007–2009 global financial crisis highlighted the need for more robust risk evaluation tools, leading to the adoption [...] Read more.

Expectiles were introduced to statistics around 40 years ago, but have recently gained renewed interest due to their relevance in financial risk management. In particular, the 2007–2009 global financial crisis highlighted the need for more robust risk evaluation tools, leading to the adoption of inference methods for expectiles. While first-order asymptotic inference results for expectiles are well established, higher-order asymptotic results remain underdeveloped. This study aims to fill that gap by deriving higher-order asymptotic results for expectiles, ultimately improving the accuracy of confidence intervals. The paper outlines the derivation of the Edgeworth expansion for both the standardized and studentized versions of the kernel-based estimator of the expectile, using large deviation results on U-statistics. The expansion is then inverted to construct more precise confidence intervals for the expectile. These theoretical results were applied to moderate sample sizes ranging from 20 to 200. To demonstrate the advantages of this methodology, an example from risk management is presented. The enhanced confidence intervals consistently outperformed those based on the first-order normal approximation. The methodology introduced in this paper can also be extended to other contexts. Full article

(This article belongs to the Special Issue Nonparametric and Semiparametric Approaches in Statistical Inference and Data Science: Theory, Methods and Applications)

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20 pages, 477 KiB

Open AccessFeature PaperArticle

Optimal Design of Multi-Asset Options

by Alejandro Balbás, Beatriz Balbás and Raquel Balbás

Risks 2025, 13(1), 16; https://doi.org/10.3390/risks13010016 - 16 Jan 2025

Cited by 1 | Viewed by 1031

Abstract

The combination of stochastic derivative pricing models and downside risk measures often leads to the paradox (risk, return) = (−infinity, +infinity) in a portfolio choice problem. The construction of a portfolio of derivatives with high expected returns and very negative downside risk (henceforth [...] Read more.

The combination of stochastic derivative pricing models and downside risk measures often leads to the paradox (risk, return) = (−infinity, +infinity) in a portfolio choice problem. The construction of a portfolio of derivatives with high expected returns and very negative downside risk (henceforth “golden strategy”) has only been studied if all the involved derivatives have the same underlying asset. This paper also considers multi-asset derivatives, gives practical methods to build multi-asset golden strategies for both the expected shortfall and the expectile risk measure, and shows that the use of multi-asset options makes the performance of the obtained golden strategy more efficient. Practical rules are given under the Black–Scholes–Merton multi-dimensional pricing model. Full article

17 pages, 396 KiB

Open AccessArticle

Recursive Estimation of the Expectile-Based Shortfall in Functional Ergodic Time Series

by Fatimah A. Almulhim, Mohammed B. Alamari, Mustapha Rachdi and Ali Laksaci

Mathematics 2024, 12(24), 3956; https://doi.org/10.3390/math12243956 - 16 Dec 2024

Viewed by 827

Abstract

This paper considers the Recursive Kernel Estimator (RKE) of the expectile-based conditional shortfall. The estimator is constructed under a functional structure based on the ergodicity assumption. More preciously, we assume that the input-variable is valued in a pseudo-metric space, output-variable is scalar and [...] Read more.

This paper considers the Recursive Kernel Estimator (RKE) of the expectile-based conditional shortfall. The estimator is constructed under a functional structure based on the ergodicity assumption. More preciously, we assume that the input-variable is valued in a pseudo-metric space, output-variable is scalar and both are sampled from ergodic functional time series data. We establish the complete convergence rate of the RKE-estimator of the considered functional shortfall model using standard assumptions. We point out that the ergodicity assumption constitutes a relevant alternative structure to the mixing time series dependency. Thus, the results of this paper allows to cover a large class of functional time series for which the mixing assumption is failed to check. Moreover, the obtained results is established in a general way, allowing to particularize this convergence rate for many special situations including the kernel method, the independence case and the multivariate case. Finally, a simulation study is carried out to illustrate the finite sample performance of the RKE-estimator. In order to examine the feasibility of the recursive estimator in practice we consider a real data example based on financial time series data. Full article

(This article belongs to the Special Issue Advances in High-Dimensional Statistics)

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21 pages, 403 KiB

Open AccessArticle

Spatio-Functional Nadaraya–Watson Estimator of the Expectile Shortfall Regression

by Mohammed B. Alamari, Fatimah A. Almulhim, Zoulikha Kaid and Ali Laksaci

Axioms 2024, 13(10), 678; https://doi.org/10.3390/axioms13100678 - 30 Sep 2024

Viewed by 853

Abstract

The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of [...] Read more.

The main aim of this paper is to consider a new risk metric that permits taking into account the spatial interactions of data. The considered risk metric explores the spatial tail-expectation of the data. Indeed, it is obtained by combining the ideas of expected shortfall regression with an expectile risk model. A spatio-functional Nadaraya–Watson estimator of the studied metric risk is constructed. The main asymptotic results of this work are the establishment of almost complete convergence under a mixed spatial structure. The claimed asymptotic result is obtained under standard assumptions covering the double functionality of the model as well as the data. The impact of the spatial interaction of the data in the proposed risk metric is evaluated using simulated data. A real experiment was conducted to measure the feasibility of the Spatio-Functional Expectile Shortfall Regression (SFESR) in practice. Full article

(This article belongs to the Special Issue Advances in Functional and Topological Data Analysis)

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19 pages, 1097 KiB

Open AccessArticle

Nonparametric Expectile Shortfall Regression for Complex Functional Structure

by Mohammed B. Alamari, Fatimah A. Almulhim, Zoulikha Kaid and Ali Laksaci

Entropy 2024, 26(9), 798; https://doi.org/10.3390/e26090798 - 18 Sep 2024

Viewed by 914

Abstract

This paper treats the problem of risk management through a new conditional expected shortfall function. The new risk metric is defined by the expectile as the shortfall threshold. A nonparametric estimator based on the Nadaraya–Watson approach is constructed. The asymptotic property of the [...] Read more.

This paper treats the problem of risk management through a new conditional expected shortfall function. The new risk metric is defined by the expectile as the shortfall threshold. A nonparametric estimator based on the Nadaraya–Watson approach is constructed. The asymptotic property of the constructed estimator is established using a functional time-series structure. We adopt some concentration inequalities to fit this complex structure and to precisely determine the convergence rate of the estimator. The easy implantation of the new risk metric is shown through real and simulated data. Specifically, we show the feasibility of the new model as a risk tool by examining its sensitivity to the fluctuation in financial time-series data. Finally, a comparative study between the new shortfall and the standard one is conducted using real data. Full article

(This article belongs to the Section Complexity)

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19 pages, 420 KiB

Open AccessArticle

k-Nearest Neighbors Estimator for Functional Asymmetry Shortfall Regression

by Mohammed B. Alamari, Fatimah A. Almulhim, Zoulikha Kaid and Ali Laksaci

Symmetry 2024, 16(7), 928; https://doi.org/10.3390/sym16070928 - 19 Jul 2024

Cited by 1 | Viewed by 1299

Abstract

This paper deals with the problem of financial risk management using a new expected shortfall regression. The latter is based on the expectile model for financial risk-threshold. Unlike the VaR model, the expectile threshold is constructed by an asymmetric least square loss function. [...] Read more.

This paper deals with the problem of financial risk management using a new expected shortfall regression. The latter is based on the expectile model for financial risk-threshold. Unlike the VaR model, the expectile threshold is constructed by an asymmetric least square loss function. We construct an estimator of this new model using the k-nearest neighbors (kNN) smoothing approach. The mathematical properties of the constructed estimator are stated through the establishment of the pointwise complete convergence. Additionally, we prove that the constructed estimator is uniformly consistent over the nearest neighbors (UCNN). Such asymptotic results constitute a good mathematical support of the proposed financial risk process. Thus, we examine the easy implantation of this process through an artificial and real data. Our empirical analysis confirms the superiority of the kNN-approach over the kernel method as well as the superiority of the expectile over the quantile in financial risk analysis. Full article

(This article belongs to the Section Mathematics)

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17 pages, 870 KiB

Open AccessArticle

Downside Risk in Australian and Japanese Stock Markets: Evidence Based on the Expectile Regression

by Kohei Marumo and Steven Li

J. Risk Financial Manag. 2024, 17(5), 189; https://doi.org/10.3390/jrfm17050189 - 2 May 2024

Viewed by 1896

Abstract

The expectile-based Value at Risk (EVaR) has gained popularity as it is more sensitive to the magnitude of extreme losses than the conventional quantile-based VaR (QVaR). This paper applies the expectile regression approach to evaluate the EVaR of stock market indices of Australia [...] Read more.

The expectile-based Value at Risk (EVaR) has gained popularity as it is more sensitive to the magnitude of extreme losses than the conventional quantile-based VaR (QVaR). This paper applies the expectile regression approach to evaluate the EVaR of stock market indices of Australia and Japan. We use an expectile regression model that considers lagged returns and common risk factors to calculate the EVaR for each stock market and to evaluate the interdependence of downside risk between the two markets. Our findings suggest that both Australian and Japanese stock markets are affected by their past development and the international stock markets. Additionally, ASX 200 index has significant impact on Nikkei 225 in terms of downside tail risk, while the impact of Nikkei 225 on ASX is not significant. Full article

(This article belongs to the Special Issue Editorial Board Members’ Collection Series: Journal of Risk and Financial Management)

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21 pages, 557 KiB

Open AccessFeature PaperArticle

Bidual Representation of Expectiles

by Alejandro Balbás, Beatriz Balbás, Raquel Balbás and Jean-Philippe Charron

Risks 2023, 11(12), 220; https://doi.org/10.3390/risks11120220 - 15 Dec 2023

Cited by 4 | Viewed by 2020

Abstract

Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing [...] Read more.

Downside risk measures play a very interesting role in risk management problems. In particular, the value at risk (VaR) and the conditional value at risk (CVaR) have become very important instruments to address problems such as risk optimization, capital requirements, portfolio selection, pricing and hedging issues, risk transference, risk sharing, etc. In contrast, expectile risk measures are not as widely used, even though they are both coherent and elicitable. This paper addresses the bidual representation of expectiles in order to prove further important properties of these risk measures. Indeed, the bidual representation of expectiles enables us to estimate and optimize them by linear programming methods, deal with optimization problems involving expectile-linked constraints, relate expectiles with VaR and CVaR by means of both equalities and inequalities, give VaR and CVaR hyperbolic upper bounds beyond the level of confidence, and analyze whether co-monotonic additivity holds for expectiles. Illustrative applications are presented. Full article

(This article belongs to the Special Issue Optimal Investment and Risk Management)

21 pages, 1000 KiB

Open AccessArticle

Spatio-Functional Local Linear Asymmetric Least Square Regression Estimation: Application for Spatial Prediction of COVID-19 Propagation

by Ali Laksaci, Salim Bouzebda, Fatimah Alshahrani, Ouahiba Litimein and Boubaker Mechab

Symmetry 2023, 15(12), 2108; https://doi.org/10.3390/sym15122108 - 23 Nov 2023

Cited by 1 | Viewed by 1116

Abstract

The problem of estimating the spatio-functional expectile regression for a given spatial mixing structure

(X_{i}, Y_{i}) \in F \times R

, when

i \in Z^{N}, N \geq 1

and

F

is a metric space, is investigated. We have [...] Read more.

The problem of estimating the spatio-functional expectile regression for a given spatial mixing structure

(X_{i}, Y_{i}) \in F \times R

, when

i \in Z^{N}, N \geq 1

and

F

is a metric space, is investigated. We have proposed the M-estimation procedure to construct the Spatial Local Linear (SLL) estimator of the expectile regression function. The main contribution of this study is the establishment of the asymptotic properties of the SLL expectile regression estimator. Precisely, we establish the almost-complete convergence with rate. This result is proven under some mild conditions on the model in the mixing framework. The implementation of the SLL estimator is evaluated using an empirical investigation. A COVID-19 data application is performed, allowing this work to highlight the substantial superiority of the SLL-expectile over SLL-quantile in risk exploration. Full article

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