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)}.
We provide a variety of results for (quasi)convex, law-invariant functionals defined on a general Orlicz space, which extend well-known results in the setting of bounded random variables. First, we show that Delbaen's representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can be always achieved under the assumption of lawinvariance. Second, we identify the range of Orlicz spaces where the characterization of the Fatou property in terms of norm lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka's representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipović and Svindland by replacing norm lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.
We introduce a class of quantile-based risk measures that generalize Value at Risk (VaR) and, likewise Expected Shortfall (ES), take into account both the frequency and the severity of losses. Under VaR a single confidence level is assigned regardless of the size of potential losses. We allow for a range of confidence levels that depend on the loss magnitude. The key ingredient is a benchmark loss distribution (BLD), that is, a function that associates to each potential loss a maximal acceptable probability of occurrence. The corresponding risk measure, called Loss VaR (LVaR), determines the minimal capital injection that is required to align the loss distribution of a risky position to the target BLD. By design, one has full flexibility in the choice of the BLD profile and, therefore, in the range of relevant quantiles. Special attention is given to piecewise constant functions and to tail distributions of benchmark random losses, in which case the acceptability condition imposed by the BLD boils down to first-order stochastic dominance. We investigate the main theoretical properties of LVaR with a focus on their comparison with VaR and ES and discuss applications to capital adequacy, portfolio risk management, and catastrophic risk.
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