Dual representation of expectile-based expected shortfall and its properties

Mekonnen Tadese; Samuel Drapeau

doi:10.3934/puqr.2021005

Article Contents

2021, Volume 6, Issue 2: 99-116. Doi: 10.3934/puqr.2021005

This issue Previous Article Next Article An infinite-dimensional model of liquidity in financial markets

Dual representation of expectile-based expected shortfall and its properties

Mekonnen Tadese^1, , and
Samuel Drapeau^2,

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
School of Mathematical Sciences & Shanghai Advanced Institute of Finance (CAFR), Shanghai Jiao Tong University, Shanghai 200030, China

Author Bio: mekonnenta@sjtu.edu.cn; sdrapeau@saif.sjtu.edu.cn
Mekonnen Tadese mekonnenta@sjtu.edu.cn
Mekonnen Tadese mekonnenta@sjtu.edu.cn

Received: November 18, 2020

Accepted: April 19, 2021

Published: June 2021

This research is supported by National Science Foundation of China (Grant No. 11971310, 11671257); “Assessment of Risk and Uncertainty in Finance” (Grant No. AF0710020) from Shanghai Jiao Tong University.

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Abstract

An expectile can be considered a generalization of a quantile. While expected shortfall is a quantile-based risk measure, we study its counterpart—the expectile-based expected shortfall—where expectile takes the place of a quantile. We provide its dual representation in terms of a Bochner integral. Among other properties, we show that it is bounded from below in terms of the convex combination of expected shortfalls, and also from above by the smallest law invariant, coherent, and comonotonic risk measures, for which we give the explicit formulation of the corresponding distortion function. As a benchmark to the industry standard expected shortfall, we further provide its comparative asymptotic behavior in terms of extreme value distributions. Based on these results, we finally explicitly compute the expectile-based expected shortfall for selected classes of distributions.

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Mathematics Subject Classification: 62P05; 91B05; 91B16; 91G70.

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Figure 1. Graphs of $ es_\alpha $ , $ tce_\alpha $ , and $ ES_\alpha $ for Pareto, standard Student $ t $ , standard normal, and beta distributions.

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Figure 2. Graph of $ \gamma_\beta $ for $ \alpha = 70 \%, \alpha = 80 \% $ , and $ \alpha = 90 \% $ .

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Figure 3. Graph of $ \varphi $ and $ \varphi_{\lambda,\beta,\delta} $ for $ \alpha = 90 \%, \lambda = 20 \% $ , $ \beta = 90 \% $ , and $ \delta = 0 \% $ .

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Figure 4. Graph of $ ES_\alpha/es_\alpha $ (for Pareto, standard Student $ t $ , and standard normal distributions) and $ (\hat{x}-ES_\alpha)/(\hat{x}-es_\alpha) $ (for the beta distribution).

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