# American Institute of Mathematical Sciences

June  2021, 6(2): 99-116. doi: 10.3934/puqr.2021005

## Dual representation of expectile-based expected shortfall and its properties

 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

Received  November 18, 2020 Accepted  April 19, 2021 Published  June 2021

Fund Project: 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.

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.

Citation: Mekonnen Tadese, Samuel Drapeau. Dual representation of expectile-based expected shortfall and its properties. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 99-116. doi: 10.3934/puqr.2021005
##### References:

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1When the measure μ is finite, most literature defines strong measurability in terms of finite-valued functions. In this particular situation, this definition of measurability is equivalent to the current definition of measurability, see Hille and Phillips [16], for instance.

2We say $L_1$ and $L_2$ in $L^1$ are comonotone, if $( L_1(\omega)-L_1(\omega'))( L_2(\omega)-L_2(\omega'))\geq 0$ for all $(\omega, \omega')\in \Omega\times \Omega$ .

3We say $F_L$ is attracted by an extreme value distribution function $H$ and denoted by $MDA(H)$ if there exist constants $c_n>0$ and $d_n\in \mathbb{R}$ for each $n$ in $\mathbb{N}$ such that　　　　　　　　　　　　　　　　　　　　 $\lim_{n\nearrow \infty} F_L^n(c_n x+d_n) = H(x).$

4A measurable function $r: \mathbb{R} \to \mathbb{R}$ is said to be slowly varying if $\lim_{t\nearrow \infty}\frac{r(tx)}{r(t)} = 1$ for each $x$ in $\mathbb{R}$ .

5 ${\cal{W}}$ is a function such that $xe^x = y$ if and only if $x = {\cal{W}}(y)$ .

##### References:
Graphs of $es_\alpha$ , $tce_\alpha$ , and $ES_\alpha$ for Pareto, standard Student $t$ , standard normal, and beta distributions.
Graph of $\gamma_\beta$ for $\alpha = 70 \%, \alpha = 80 \%$ , and $\alpha = 90 \%$ .
Graph of $\varphi$ and $\varphi_{\lambda,\beta,\delta}$ for $\alpha = 90 \%, \lambda = 20 \%$ , $\beta = 90 \%$ , and $\delta = 0 \%$ .
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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