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In the\ud present contribution, under weak\ud technical assumptions on the elicitability scoring function S we fully characterize the class of convex and\ud coherent risk measures that are elicitable with an accuracy rewarding\ud scoring function. In particular, we answer the question\ud posed by Ziegel (2013), showing that expectiles are indeed the\ud only elicitable coherent risk measure

In the present contribution, we characterise law determined convex risk measures that have convex level sets at the level of distributions. By relaxing the assumptions in Weber (Math. Finance 16:419–441, 2006), we show that these risk measures can be identified with a class of generalised shortfall risk measures. As a direct consequence, we are able to extend the results in Ziegel (Math. Finance, 2014, http://​onlinelibrary.​wiley.​com/​doi/​10.​1111/​mafi.​12080/​abstract) and Bellini and Bignozzi (Quant. Finance 15:725–733, 2014) on convex elicitable risk measures and confirm that expectiles are the only elicitable coherent risk measures. Further, we provide a simple characterisation of robustness for convex risk measures in terms of a weak notion of mixture continuity

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The notion of residual estimation risk is introduced in order to study the impact of parameter uncertainty on capital adequacy, for a given risk measure and capital estimation procedure. Residual estimation risk is derived by applying the risk measure on a portfolio consisting of a random loss and a capital estimator, reflecting the randomness inherent in the data. Residual risk thus equals the additional amount of capital that needs to be added to the portfolio to make it acceptable. We propose modified capital estimation procedures, based on parametric bootstrapping and on predictive distributions, which tend to increase capital requirements, by compensating for parameter uncertainty and leading to a residual risk close to zero. In the particular case of location-scale families of distributions, the analysis simplifies substantially and a capital estimator can always be found that leads to a residual risk of exactly zero.