Expectile risk quadrangles and applications

This paper studies the expectile risk measure within the fundamental risk quadrangle (FRQ) framework. We rigorously examine the properties of constructed expectile quadrangles, with particular emphasis on a new quadrangle where the expectile is both a statistic and a risk measure. This expectile quadrangle is based on a new piecewise linear error, which is an alternative to the standard asymmetric mean squared error. We demonstrate the equivalence of various expectile regressions with these errors. Linear regression with the piecewise linear error is reduced to linear programming. The optimization of the expectile is reduced to convex and linear programming using the FRQ regret theorem. The Kusuoka representation of the expectile implies the equivalence of conditional value-at-risk and expectile portfolio optimization. It is known that omega and expectile portfolio optimization are equivalent. Therefore, conditional value-at-risk, expectile and omega portfolio optimization are equivalent. Finally, the FRQ regression theorem implies the equivalence of expectile and quantile regressions. The theoretical findings of the paper are validated with several case studies.