[Paper Review] Set-Valued Dynamic Risk Measures

Zachary Feinstein, Birgit Rudloff|arXiv (Cornell University)|Jan 6, 2012

Risk and Portfolio Optimization2 citations

TL;DR

This paper introduces multi-portfolio time consistency as a stronger, more appropriate generalization of time consistency for set-valued dynamic risk measures in markets with transaction costs. It establishes that this property is equivalent to both a recursive formulation and an additive structure in acceptance sets, unlike in the scalar case where time consistency and recursion coincide.

ABSTRACT

The paper concerns primal and dual representations as well as time consistency of set-valued dynamic risk measures. Set-valued risk measures appear naturally when markets with transaction costs are considered and capital requirements can be made in a basket of currencies or assets. Time consistency of scalar risk measures can be generalized to set-valued risk measures in different ways. The most intuitive generalization is called time consistency. We will show that the equivalence between a recursive form of the risk measure and time consistency, which is a central result in the scalar case, does not hold in the set-valued framework. Instead, we propose an alternative generalization, which we will call multi-portfolio time consistency and show in the main result of the paper that this property is indeed equivalent to the recursive form as well as to an additive property for the acceptance sets. Multi-portfolio time consistency is a stronger property than time consistency. In the scalar case, both notions coincide.

Motivation & Objective

• To generalize time consistency from scalar to set-valued risk measures in the context of financial markets with transaction costs.
• To address the failure of the classical equivalence between recursive formulation and time consistency in the set-valued framework.
• To propose and formalize a stronger, more suitable generalization: multi-portfolio time consistency.
• To establish the equivalence between multi-portfolio time consistency, recursive representation, and additivity of acceptance sets.

Proposed method

• Introduce set-valued risk measures as capital requirements in baskets of assets or currencies, reflecting transaction costs.
• Define time consistency in the set-valued context and show its insufficiency in preserving the recursive structure.
• Propose multi-portfolio time consistency as a stronger alternative, ensuring consistency across portfolios over time.
• Formulate the recursive representation of risk measures and prove its equivalence to multi-portfolio time consistency.
• Characterize acceptance sets as additive under multi-portfolio time consistency, linking the property to set structure.
• Use duality theory to derive primal and dual representations of the risk measures, supporting the theoretical framework.

Experimental results

Research questions

• RQ1 How can time consistency be meaningfully generalized in the set-valued framework where the classical equivalence to recursion fails?
• RQ2 What alternative property ensures consistency in dynamic set-valued risk measurement under transaction costs?
• RQ3 Is there a formulation of set-valued risk measures that preserves the recursive structure seen in the scalar case?
• RQ4 How do the acceptance sets of set-valued risk measures behave under the proposed time consistency concept?
• RQ5 What is the relationship between multi-portfolio time consistency, recursive representation, and additivity of acceptance sets?

Key findings

• Multi-portfolio time consistency is a stronger property than standard time consistency in the set-valued setting, where the two notions do not coincide.
• The proposed multi-portfolio time consistency is equivalent to the recursive form of the risk measure, restoring the classical equivalence lost in the set-valued case.
• Acceptance sets under multi-portfolio time consistency exhibit an additive property, meaning the union of sets at different times preserves the risk measure's structure.
• The primal and dual representations of set-valued risk measures are derived, supporting the theoretical foundation of the framework.
• The framework successfully generalizes scalar risk measure properties to markets with transaction costs, where multiple assets or currencies are used for capital requirements.