Beyond Cash-Additive Capital Requirements: When Changing the Numeraire Fails

Abstract:     
We discuss general capital requirements representing the minimum amount of capital that a financial institution needs to raise and invest in a pre-specified eligible, or reference, asset to ensure it is adequately capitalized. Financial positions are modeled as elements belonging to an ordered topological vector space. The payoff of the eligible asset is assumed to be an arbitrary non-zero positive element, thus allowing for a wide range of choices. In the context of function spaces, these general capital requirements cannot be transformed into cash-additive capital requirements by a simple change of numeraire unless the payoff of the eligible asset is bounded away from zero. This excludes the possibility of choosing a defaultable security as the eligible asset which, given the potential unavailability of risk-free assets, constitutes an important gap in the existing theory of capital requirements. This paper fills this gap and provides a detailed analysis of the interplay between acceptance sets and eligible assets. We provide a variety of finiteness and continuity results when the eligible asset has a payoff with "interior-like" qualities, paying particular attention to the case where the underlying space of positions is a Fréchet lattice. As an application we provide a complete characterization of finiteness and L^p-continuity for quantile-based capital requirements, the most important types of capital requirements encountered in practice.

Number of Pages in PDF File: 32

Keywords: ordered topological vector spaces, Fréchet and Banach lattices, acceptance sets, single eligible asset, risk measures, capital adequacy, value-at-risk, tail value-at-risk

JEL Classification: C65, D81, G22

working papers series