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Expectile bears some interesting properties in comparison to the industrywide expected shortfall in terms of assessment of tail risk. We study therelationship between expectile and expected shortfall using duality results andthe link to optimized certainty equivalent. Lower and upper bounds of expectileare derived in terms of expected shortfall as well as a characterization ofexpectile in terms of expected shortfall. Further, we study the asymptoticbehavior of expectile with respect to expected shortfall as the confidencelevel goes to $1$ in terms of extreme value distributions. We use concentrationinequalities to illustrate that the estimation of value at risk requires largersample size than expected shortfall and expectile for heavy tail distributionswhen $\alpha$ is close to $1$. Illustrating the formulation of expectile interms of expected shortfall, we also provide explicit or semi-explicitexpressions of expectile and some simulation results for some classicaldistributions.

Relative Bound and Asymptotic Comparison of Expectile with Respect to Expected Shortfall