  Procyclicality of historical risk measure estimation means that one tends to
over-estimate future risk when present realized volatility is high and vice
versa under-estimate future risk when the realized volatility is low. Out of it
different questions arise, relevant for applications and theory: What are the
factors which affect the degree of procyclicality? More specifically, how does
the choice of risk measure affect this? How does this behaviour vary with the
choice of realized volatility estimator? How do different underlying model
assumptions influence the pro-cyclical effect? In this paper we consider three
different well-known risk measures (Value-at-Risk, Expected Shortfall,
Expectile), the r-th absolute centred sample moment, for any integer $r>0$, as
realized volatility estimator (this includes the sample variance and the sample
mean absolute deviation around the sample mean) and two models (either an iid
model or an augmented GARCH($p$,$q$) model). We show that the strength of
procyclicality depends on these three factors, the choice of risk measure, the
realized volatility estimator and the model considered. But, no matter the
choices, the procyclicality will always be present.
