  This paper introduces and studies factor risk measures. While risk measures
only rely on the distribution of a loss random variable, in many cases risk
needs to be measured relative to some major factors. In this paper, we
introduce a double-argument mapping as a risk measure to assess the risk
relative to a vector of factors, called factor risk measure. The factor risk
measure only depends on the joint distribution of the risk and the factors. A
set of natural axioms are discussed, and particularly distortion, quantile,
linear and coherent factor risk measures are introduced and characterized.
Moreover, we introduce a large set of concrete factor risk measures and many of
them are new to the literature, which are interpreted in the context of
regulatory capital requirement. Finally, the distortion factor risk measures
are applied in the risk-sharing problem and some numerical examples are
presented to show the difference between the Value-at-Risk and the quantile
factor risk measures.
