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. DOI: https://doi.org/10.1007/s11579-014-0118-0 Posted at the Zurich Open Repository and Archive, University of Zurich ZORA URL: https://doi.org/10.5167/uzh-119589 Accepted Version Originally published at: Farkas, Walter; Koch-Medina, Pablo; Munari, Cosimo-Andrea (2015). Measuring risk with multiple eligible assets. Mathematics and Financial Economics, 9(1):3-27. DOI: https://doi.org/10.1007/s11579-014-0118-0 Noname manuscript No. (will be inserted by the editor) Measuring risk with multiple eligible assets Walter Farkas · Pablo Koch-Medina · Cosimo Munari Received: / Accepted: Abstract 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 character-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 characterPartial support through the SNF project 51NF40-144611 “Capital adequacy, valuation, and portfolio selection for insurance companies” is gratefully acknowledged. Walter Farkas University of Zurich and ETH Zurich Department of Banking and Finance Plattenstrasse 14, 8032 Zurich, Switzerland E-mail: [email protected] Pablo Koch-Medina University of Zurich Department of Banking and Finance Plattenstrasse 14, 8032 Zurich, Switzerland E-mail: [email protected] Cosimo Munari ETH Zurich Department of Mathematics Rämistrasse 101, 8092 Zurich, Switzerland E-mail: [email protected] 2 Walter Farkas et al. ization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.