Title: Factor risk measures

Abstract: 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.