The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

We derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems. This result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation. In particular, we characterize the bifurcation values from infinity to positive steady states. The stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed.

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.

This paper is concerned with clarifying the link between risk measurement and capital efficiency. For this purpose we introduce risk measurement as the minimum cost of making a position acceptable by adding an optimal combination of multiple eligible assets. Under certain assumptions, it is shown that these risk measures have properties similar to those of coherent risk measures. The motivation for this paper was the study of a multi-currency setting where it is natural to use simultaneously a domestic and a foreign asset as investment vehicles to inject the capital necessary to make an unacceptable position acceptable. We also study what happens when one changes the unit of account from domestic to foreign currency and are led to the notion of compatibility of risk measures. In addition, we aim to clarify terminology in the field.

In a capital adequacy framework, risk measures are used to determine the minimal amount of capital that a financial institution has to raise and invest in a portfolio of pre-specified eligible assets in order to pass a given capital adequacy test. From a capital efficiency perspective, it is important to identify the set of portfolios of eligible assets that allow to pass the test by raising the least amount of capital. We study the existence and uniqueness of such optimal portfolios as well as their sensitivity to changes in the underlying capital position. This naturally leads to investigating the continuity properties of the set-valued map associating to each capital position the corresponding set of optimal portfolios. We pay special attention to lower semicontinuity, which is the key continuity property from a financial perspective. This "stability" property is always satisfied if the test is based on a polyhedral risk measure but it generally fails once we depart from polyhedrality even when the reference risk measure is convex. However, lower semicontinuity can be often achieved if one if one is willing to focuses on portfolios that are close to being optimal. Besides capital adequacy, our results have a variety of natural applications to pricing, hedging, and capital allocation problems.The minimal amount of capital that needs to be raised and invested in a portfolio of eligible payoffs to pass the prescribed acceptability test is represented by the risk measure ρ : X → R defined byBy definition, the quantity ρ(X) admits a natural operational interpretation as a capital requirement. The risk measures introduced by Artzner et al. (1999) constitute the prototype of the above capital requirement functionals and correspond to the simplest form of management action, i.e. raising capital and investing it in a single eligible asset. The extension to a multi-asset framework was first dealt with, to the best of our knowledge, in Föllmer and Schied (2002) and later taken up in Frittelli and Scandolo (2006), Artzner et al. (2009) and in the more comprehensive study by Farkas et al. (2015). Optimal eligible payoffsThis paper is concerned with the study of the set-valued map E : X ⇒ M defined by E(X) = {Z ∈ M ; X + Z ∈ A, π(Z) = ρ(X)}.