On Fairness of Systemic Risk Measures

TL;DR

This paper advances the understanding of systemic risk measures by establishing dual representations, optimal allocations, and a fair risk allocation framework, with explicit results for exponential utilities.

Contribution

It proves dual representations and uniqueness of optimal allocations for a subclass of systemic risk measures, and introduces a utility maximization approach linked to fair risk allocation.

Findings

Dual representation of systemic risk measures established

Existence and uniqueness of optimal allocations proven

Explicit solutions provided for exponential utility case

Abstract

In our previous paper, "A Unified Approach to Systemic Risk Measures via Acceptance Set" (\textit{Mathematical Finance, 2018}), we have introduced a general class of systemic risk measures that allow for random allocations to individual banks before aggregation of their risks. In the present paper, we prove the dual representation of a particular subclass of such systemic risk measures and the existence and uniqueness of the optimal allocation related to them. We also introduce an associated utility maximization problem which has the same optimal solution as the systemic risk measure. In addition, the optimizer in the dual formulation provides a \textit{risk allocation} which is fair from the point of view of the individual financial institutions. The case with exponential utilities which allows for explicit computation is treated in details.