Towards a unified theory of return risk measures

Classical risk measures are designed to evaluate the risk of uncertain monetary payoffs (or losses), whereas in time series analysis, risk is typically assessed for logarithmic returns. An axiomatic foundation for this latter perspective has recently been developed under the umbrella of so-called return risk measures (RRMs). In this theory, the positive cone of the linear space of essentially bounded random variables, or subsets of this cone, such as its interior, plays a key role.

Our contribution consists of extending the RRM definition to general ordered vector spaces and characterizing positive homogeneity in terms of the geometric epigraph. Then, to study geometric convexity and establish connections to classical risk measures, we specialize to AM-algebras admitting an order unit. Geometric convexity is also characterized via the geometric epigraph. This setting encompasses a variety of domains, including Euclidean spaces and spaces of multidimensional and essentially bounded random variables. The latter is new in the RRM literature and enables us to study several novel classes of RRMs, including multivariate RRMs, systemic RRMs, and their set-valued versions. We present results regarding standard properties; for example, finiteness, separability and dual representations. Finally, we introduce and discuss concrete kinds of multivariate RRMs and illustrate their performances numerically.