90 research outputs found
Measuring risk with multiple eligible assets
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
Beyond cash-additive risk measures: when changing the num\'{e}raire fails
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 numeraire. 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 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
Capital adequacy tests and limited liability of financial institutions
The theory of acceptance sets and their associated risk measures plays a key
role in the design of capital adequacy tests. The objective of this paper is to
investigate, in the context of bounded financial positions, the class of
surplus-invariant acceptance sets. These are characterized by the fact that
acceptability does not depend on the positive part, or surplus, of a capital
position. We argue that surplus invariance is a reasonable requirement from a
regulatory perspective, because it focuses on the interests of liability
holders of a financial institution. We provide a dual characterization of
surplus-invariant, convex acceptance sets, and show that the combination of
surplus invariance and coherence leads to a narrow range of capital adequacy
tests, essentially limited to scenario-based tests. Finally, we emphasize the
advantages of dealing with surplus-invariant acceptance sets as the primary
object rather than directly with risk measures, such as loss-based and
excess-invariant risk measures, which have been recently studied by Cont,
Deguest, and He (2013) and by Staum (2013), respectively
Qualitative robustness of utility-based risk measures
We contribute to the literature on statistical robustness of risk measures by computing the index of qualitative robustness for risk measures based on utility functions. This problem is intimately related to finding the natural domain of finiteness and continuity of such risk measures
Risk Measures and Efficient use of Capital
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 fiel
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