Risk Measures and Capital Requirements with Multiple Eligible Assets

Abstract:     
We provide a comprehensive treatment of risk measures with respect to multiple eligible assets in the context general ordered topological vector spaces. The adequacy of the capitalization of a financial institution is typically defined in terms of acceptance sets of financial positions. Risk measures are used to determine the minimum amount of capital - the so-called capital requirement - that has to be raised and invested in a portfolio of a prespecified class of tradable assets - the so-called eligible assets - to make a position acceptable.

We investigate when capital requirements are finitely valued, when they are a continuous function of financial positions, and when two capital requirements coincide. We show how to reduce risk measures with respect to multiple eligible assets to the risk measures with respect to a single eligible asset by properly enlarging the acceptance set. Risk measures with respect to multiple eligible assets are shown to be non-trivial when no acceptability arbitrage is possible, i.e. when not every position can be made acceptable by adding a zero-cost portfolio of eligible assets. We derive a theorem on the structure of closed convex acceptance sets based solely on the external characterization of general closed convex sets. This theorem provides us with a simple, new approach to obtaining dual representations of convex risk measures with respect to multiple eligible assets. Our results also generalize well-known results in the single-eligible-asset case. A distinguishing feature of our approach is that convex risk measures are represented as the supremum of an objective function that depends exclusively on the acceptance set, where the supremum is taken over a set that varies with the choice of the class of eligible assets.

Number of Pages in PDF File: 28

Keywords: acceptance sets, multiple eligible assets, capital adequacy, risk measures, Value-at-Risk, Tail Value-at-Risk, acceptability arbitrage

JEL Classification: C60, G11, G22

working papers series