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Measuring risk with multiple eligible assets

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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 characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

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Notes

  1. For our purposes it suffices to know that there is a linear pricing functional assigning market values to marketed payoffs. For more details on the underlying market models, also for the case of infinite dimensional marketed spaces, we refer to Clark [8] and Kreps [21].

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Acknowledgments

Financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management” (NCCR FinRisk), project “Mathematical Methods in Financial Risk Management”, is gratefully acknowledged by W. Farkas and C. Munari. Part of this research was supported by Swiss Re.

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Authors and Affiliations

  1. Department of Banking and Finance, University of Zurich and ETH Zurich, Plattenstrasse 14, 8032,  Zurich, Switzerland

    Walter Farkas

  2. Department of Banking and Finance, University of Zurich, Plattenstrasse 14, 8032,  Zurich, Switzerland

    Pablo Koch-Medina

  3. Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092,  Zurich, Switzerland

    Cosimo Munari

Authors
  1. Walter Farkas
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  2. Pablo Koch-Medina
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  3. Cosimo Munari
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Corresponding author

Correspondence to Pablo Koch-Medina.

Additional information

Partial support through the SNF project 51NF40-144611 “Capital adequacy, valuation, and portfolio selection for insurance companies” is gratefully acknowledged.

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Farkas, W., Koch-Medina, P. & Munari, C. Measuring risk with multiple eligible assets. Math Finan Econ 9, 3–27 (2015). https://doi.org/10.1007/s11579-014-0118-0

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