[Paper Review] Valuations and dynamic convex risk measures

Arnaud Jobert, L. C. G. Rogers|RePEc: Research Papers in Economics|Sep 3, 2007

Risk and Portfolio Optimization26 references4 citations

TL;DR

This paper introduces a dynamic convex risk measurement framework using concave valuation operators that satisfy axioms of concavity, positivity, monotonicity, and time consistency. It establishes that risk-sharing among subsidiaries and market access preserve these axioms, ensuring stable, time-consistent valuations through a dual representation involving penalty functions and state-price densities.

ABSTRACT

This paper approaches the definition and properties of dynamic convex risk measures through the notion of a family of concave valuation operators satisfying certain simple and credible axioms. Exploring these in the simplest context of a finite time set and finite sample space, we find natural risk-transfer and time-consistency properties for a firm seeking to spread its risk across a group of subsidiaries.

Motivation & Objective

• To develop a dynamic framework for convex risk measurement that respects time evolution and risk transfer across subsidiaries.
• To formalize the concept of dynamic convex risk measures through valuation operators rather than acceptance sets, emphasizing time consistency and risk-sharing.
• To show that risk-sharing among subsidiaries leads to valuations that satisfy the same axioms as the original, ensuring long-term stability.
• To extend the framework to include access to financial markets, demonstrating that market participation yields consistent, modified valuations.
• To establish a dual representation for dynamic valuations using penalty functions and state-price density processes.

Proposed method

• Define valuation operators π as the negative of risk measures, satisfying axioms: concavity, positive homogeneity (in the coherent case), monotonicity, and translation invariance.
• Use a finite time and finite sample space setting to simplify analysis and derive properties of dynamic valuations.
• Establish time consistency via a 'pasting' property: π_tT(Y) = E_t[ζ_T Y]/ζ_t, where ζ is a state-price density process.
• Represent dynamic valuations via dual forms: π(X) = inf_{Q∈Q} {E_Q[X] - α(Q)}, generalizing coherent risk measures.
• Decompose multi-period valuations into one-period components, enabling recursive computation and interpretation.
• Prove that risk-sharing and market access preserve the axiomatic structure, ensuring dynamic consistency.

Experimental results

Research questions

• RQ1 How can dynamic convex risk measures be consistently defined over time in a multi-period setting?
• RQ2 What axiomatic properties ensure time consistency and risk-transfer stability in a dynamic valuation framework?
• RQ3 How does risk-sharing among subsidiaries affect the valuation process and its adherence to convexity and time consistency?
• RQ4 What is the impact of market access on the structure of dynamic valuations and their dual representations?
• RQ5 Can dynamic valuations be represented via state-price densities and penalty functions, and what is the role of the Radon-Nikodym theorem in this representation?

Key findings

• Dynamic convex valuations can be represented as π(X) = inf_{Q∈Q} {E_Q[X] - α(Q)}, where α is a concave penalty function, generalizing coherent risk measures.
• Risk-sharing among subsidiaries results in valuations that satisfy the same axioms as the original, ensuring long-term consistency and stability.
• The time consistency of the valuation system is preserved under risk-sharing, meaning the firm’s optimal risk allocation remains optimal at all future times.
• Market access provides a fixed benefit to the firm, but the resulting valuations still satisfy the same axiomatic framework, now with modified penalty functions.
• The dynamic valuation operators admit a dual representation via state-price density processes: π_tT(Y) = E_t[ζ_T Y]/ζ_t, linking to standard risk-neutral pricing.
• The Radon-Nikodym theorem ensures the existence of a state-price density process ζ_T such that π_0T(Y) = E[ζ_T Y], establishing a probabilistic foundation for the valuation.