Beyond cash-additive risk measures: when changing the numéraire fails

Abstract

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numéraire. However, discounting does not work in all financially relevant situations, typically when the eligible asset is a defaultable bond. In this paper we fill this gap allowing for general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on Value-at-Risk and Tail Value-at-Risk on L p spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that, when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.

Portfolio Optimization with Quasiconvex Risk Measures

“…We stress that, to our knowledge, this article is the first attempt to study an optimal portfolio problem associated with quasiconvex risk measures and that no assumption on cash-subadditivity is imposed on the risk measures considered. This choice is also supported by a recent paper of Farkas et al [15], where cash-subadditivity (with or without quasiconvexity) is discussed. As emphasized by the authors, indeed, cash-subadditivity is reasonable only when the reference (or eligible) asset cannot default.…”

“…We will work with quasiconvex risk measures in full generality, not assuming cash-subadditivity of . The reason is twofold: first, we are interested in results holding for general quasiconvex risk measures; second, as remarked by Farkas et al [15], it is not always reasonable to assume cash-subadditivity (e.g., not when the reference asset is a defaultable bond).…”

A continuous selection for optimal portfolios under convex risk measures does not always exist

“…In the financial literature the above function is usually referred to as a risk measure. The interested reader can consult [2], [3], [6], [7], [8] for a variety of results on risk measures and discussions on their financial relevance in different areas of mathematical finance.…”

Model spaces for risk measures

“…U ≥ δ for some constant δ > 0. In [11,12], such securities are called non-defaultable. We will show that if the acceptance set is "nice enough", then any finite risk measure arising from it in an a priori model-free framework like L ∞ indeed implies a probabilistic model, a so-called weak reference model; see Theorem 3.3.…”