Computing expectiles via fixed point iterations

Expectiles are statistical parameters which also provide a class of sublinear financial risk measures. They are solutions of continuous optimization problems. The corresponding first order condition provides two different fixed point characterizations for expectiles, both of which can be utilized for computing them. Although especially the so-called two-sided version is already implemented and widely used, a general convergence proof appears to be new. Finite termination conditions for sample versions are also given.