Abstract: From the viewpoints of quantile functions, we gave the
definition of AVaR of financial positions under a capacity. Then,
using the classical results of AVaR under the probability measure,
we established the representation theorem for AVaR under the
submodular capacity. As a byproduct of this representation theorem,
we proved that AVaR under a submodular capacity is a coherent risk
measure, which generalized the classical results.

Key words: AVaR, quantile function, representation theorem, submodular capacity