ASTIN Bulletin

Research Article

Risk Measures and Efficient use of Capital   1

Philippe Artznera12, Freddy Delbaena23 and Pablo Koch-Medinaa34

a1 Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS et Laboratoire de Recherches en Gestion, FR 67084 Strasbourg, France, E-mail:

a2 Departement für Mathematik, Eidgenössische Technische Hochschule, ETH-Zentrum, CH 8092 Zürich, Schweiz, E-mail:

a3 Swiss Reinsurance Company, Mythenquai 50/60, CH 8022 Zürich, Schweiz, E-mail:


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 field.


  • Acceptance set;
  • accounting;
  • currency compatibility;
  • eligible asset;
  • no acceptability arbitrage;
  • non acceptability of leverage;
  • numéraire;
  • risk-free investment;
  • risk measurement;
  • supervision


1   This paper elaborates, with a major change, on presentations to the Isaac Newton Institute, DQF Program, February 2005, the Solvency II Tagung, University of Karlsruhe, April 2005, the ASTIN/AFIR Colloquium, Zürich, September 2005 and to the DGVFM Insurance Day, Cologne, April 2006. Thanks are due to referees for insisting on the need of comments related to the setting of the paper in the current literature, and to Jean-Marc Eber and Patrice Poncet for interesting discussions.

2   Partial supports from AERF/CKER, The Actuarial Foundation and from the Isaac Newton Institute are gratefully acknowledged.

3   Partial support from Credit Suisse is gratefully acknowledged.

4   This author expresses his personal view in the paper.