Risk measurements are clearly central to risk management, in particular for banks, (re)insurance companies, and investment funds. The question of the appropriateness of risk measures for evaluating the risk of financial institutions has been heavily debated, especially after the financial crisis of 2008/2009. Another concern for financial institutions is the pro-cyclicality of risk measurements. In this paper, we extend existing work on the pro-cyclicality of the Value-at-Risk to its main competitors, Expected Shortfall, and Expectile: We compare the pro-cyclicality of historical quantile-based risk estimation, taking into account the market state. To characterise the latter, we propose various estimators of the realised volatility. Considering the family of augmented GARCH(p, q) processes (containing well-known GARCH models and iid models, as special cases), we prove that the strength of pro-cyclicality depends on the three factors: the choice of risk measure and its estimators, the realised volatility estimator and the model considered, but, no matter the choices, the pro-cyclicality is always present. We complement this theoretical analysis by performing simulation studies in the iid case and developing a case study on real data.

Complete inventories of the fauna at a given place, for a specific community or geographical area are often exceedingly hard to get. In recent years a number of estimation techniques have emerged that can be used to extrapolate from these samples to the true number of species in an area. These estimation models are based on different mathematical approaches and can be classified as either species accumulation curves or nonparametric estimators (Brose et al., 2003, Ecology, 84, 2364–2377). In this paper, we have tested the performance of some of the richness estimators on nineteen mountain ponds in Castilla y León (Spain) in order to provide guidance on their potential use in future researches. We collected benthic macroinvertebrate of these ponds from the littoral zone with a pond net by kicking and sweeping. Ten-second samples were collected in each pond up to a total time of 3 to 5 minutes per pond, depending on the pond size. In addition, two of the ponds, were sampled in 2004, 2006 and 2007 providing a three-year time series. The results of this study showed that Jackknife 2 was the best of the evaluated methods based on all chosen criteria and also performed well across all studied ponds. Jackknife 1, Chao 1 and Chao 2 also presented good results and they were inferior to Jackknife 2 mainly because of the requirement for larger sub-sample sizes.