Approaches to Conditional Risk

36 Pages Posted: 1 Feb 2011 Last revised: 16 Feb 2012

See all articles by Damir Filipović

Damir Filipović

Ecole Polytechnique Fédérale de Lausanne; Swiss Finance Institute

Michael Kupper

Humboldt University of Berlin - Department of Mathematics

Nicolas Vogelpoth

Vienna Institute of Finance

Date Written: January 13, 2011


We present and compare two different approaches to conditional risk measures. One approach draws from convex analysis in vector spaces and presents risk measures as functions on Lp spaces, while the other approach utilizes module-based convex analysis where conditional risk measures are defined on Lp type modules. Both approaches utilize general duality theory for vector valued convex functions in contrast to the current literature in which we find ad hoc dual representations. By presenting several applications such as monotone and (sub)cash invariant hulls with corresponding examples we illustrate that module-based convex analysis is well suited to the concept of conditional risk measures.

Keywords: Conditional risk measures, L0-modules, Lp type modules, Monotone hulls, Subcash invariant hulls, Cash invariant hulls

JEL Classification: C60

Suggested Citation

Filipovic, Damir and Kupper, Michael and Vogelpoth, Nicolas, Approaches to Conditional Risk (January 13, 2011). Swiss Finance Institute Research Paper No. 11-02, Available at SSRN: or

Damir Filipovic (Contact Author)

Ecole Polytechnique Fédérale de Lausanne ( email )

Station 5
Lausanne, 1015


Swiss Finance Institute

c/o University of Geneva
40, Bd du Pont-d'Arve
CH-1211 Geneva 4

Michael Kupper

Humboldt University of Berlin - Department of Mathematics ( email )

Unter den Linden
Berlin, D-10099

Nicolas Vogelpoth

Vienna Institute of Finance ( email )

Nordbergstrasse 15
Vienna, 1090

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