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A Unified Approach to Systemic Risk Measures via Acceptance Sets

Francesca Biagini, Jean-Pierre Fouque, Marco Frittelli, Thilo Meyer-Brandis

arXiv:1503.06354v2q-fin.MFq-fin.RM

TL;DR

Univariate measures applied to individual institutions do not sufficiently capture systemic risk from interconnectedness and contagion. The paper develops a flexible acceptance-set framework for systemic risk and establishes conditions for risk-measure properties including convexity and quasi-convexity.

  • Problem

    Univariate monetary risk measures for individual institutions do not sufficiently capture systemic risk generated by interconnectedness and contagion.

    Source: §1.1 · p. 5

  • Method

    The framework combines multidimensional acceptance sets and aggregation functions, allowing capital allocation before aggregation, including scenario-dependent allocations.

    Source: §1.1 · p. 7, §1.3 · p. 9

  • Results

    For the systemic risk measure in (3.2), conditions (P4) and (P5) ensure quasi-convexity, while (P4) and (P6) ensure convexity.

    Source: §3 · p. 19

  • Takeaways & Limitations

    When allocations are deterministic, the optimal allocation is unique and permits an unambiguous ranking of institutions by systemic riskiness; scenario-dependent allocations may not yield a unique ranking.

    Source: §4.1 · pp. 27–28

    Allowing capital addition before aggregation can lose quasi-convexity in general, so this approach requires nontrivial restrictions in the cases described.

    Source: §3 · p. 20

Abstract

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The financial crisis has dramatically demonstrated that the traditional approach to apply univariate monetary risk measures to single institutions does not capture sufficiently the perilous systemic risk that is generated by the interconnectedness of the system entities and the corresponding contagion effects. This has brought awareness of the urgent need for novel approaches that capture systemic riskiness. The purpose of this paper is to specify a general methodological framework that is flexible enough to cover a wide range of possibilities to design systemic risk measures via multi-dimensional acceptance sets and aggregation functions, and to study corresponding examples. Existing systemic risk measures can usually be interpreted as the minimal capital needed to secure the system after aggregating individual risks. In contrast, our approach also includes systemic risk measures that can be interpreted as the minimal capital funds that secure the aggregated system by allocating capital to the single institutions before aggregating the individual risks. This allows for a possible ranking of the institutions in terms of systemic riskiness measured by the optimal allocations. Moreover, we also allow for the possibility of allocating the funds according to the future state of the system (random allocation). We provide conditions which ensure monotonicity, convexity, or quasi-convexity properties of our systemic risk measures.

1 Introduction

The paper develops an acceptance-set framework for measuring systemic risk from institutions’ uncertain positions, using aggregation to represent reduced-form systems or contagion. It covers capital allocation before or after aggregation, including scenario-dependent allocations, and previews theoretical results and applications to Gaussian and finite-probability systems.

1 Introduction

  • The framework treats the institutions’ position vector as primitive, either summarizing a complex system’s risks or representing positions before contagion, and compares system-wide risk.It does not require a specific structural network model.Source: §1 · p. 3

1.1 From one-dimensional to N-dimensional risk profiles

  • Traditional institution-by-institution monetary risk measures miss systemic risk from interconnectedness and contagion; common alternatives first aggregate positions and then assess the resulting system risk.The paper proposes a broader framework using acceptance sets and aggregation functions.Source: §1.1 · pp. 5, 7

1.2 First add capital, then aggregate

  • Adding capital to institutions before aggregation measures the minimum total injection needed to secure the system and can rank institutions by their optimal allocations.An optimal allocation need not be unique, so the resulting ranking may require a selection criterion.Source: §1.2 · pp. 7–8

1.3 First add scenario-dependent allocation, then aggregate

  • The framework permits admissible random asset allocations whose payoffs can depend on the realized scenario, including lender-of-last-resort cash distributed at time T under restrictions specified by C.With C = R^N, distributions are determined today; with C = C^R, they may vary freely by scenario, including withdrawals.Source: §1.3 · pp. 8–9
  • Scenario-dependent allocations can make systemic risk depend on the joint distribution of institutions’ positions, even when acceptability uses only marginal distributions.With unrestricted cross-subsidization, the canonical measure instead applies a univariate risk measure to the sum of positions.Source: §1.3 · pp. 9–10

1.4 Multi-dimensional Acceptance Sets

  • A multidimensional acceptance set extends the framework to the minimum-cost admissible asset vector that makes the augmented positions acceptable as a vector.This includes componentwise acceptance sets for individual institutions, which need not be expressible as acceptance of an aggregated loss.Source: §1.4 · pp. 10–11

1.5 Degree of Acceptability

  • The most general formulation assigns an acceptance set to each admissible allocation, unifying the earlier cases and allowing aggregation rules beyond a single fixed acceptance set.Its value need not represent cash added to the system; it can represent the minimum aggregated risk level at which the system is acceptable.Source: §1.5 · p. 11
  • A joint aggregation function can represent both injecting capital before aggregation and aggregating risks before injecting capital.The broader acceptance-set dependence on capital allocation also permits multidimensional quasi-convex risk measures, but generally removes cash additivity.Source: §1.5 · pp. 11–12
  • The theoretical framework establishes conditions for well-posed systemic risk measures with decreasing monotonicity and quasi-convexity or convexity.It then examines measures based on aggregation and one-dimensional acceptability, as well as scenario-dependent allocations.Source: §1.5 · p. 12
  • The Gaussian-system example considers deterministic and selected random cash allocations, including interacting diffusions that model interinstitutional borrowing and lending.The paper also presents a finite-probability example where systemic risk measures can be computed for very general random cash allocations.Source: §1.5 · p. 12

2 Definition of Systemic Risk Measures and Properties

The section defines systemic risk through acceptable risk vectors and the cost of the adjustments that make a system acceptable. It then gives conditions under which this measure is monotone, quasi-convex, or convex.

Structural properties

  • Monotonicity follows when acceptable sets respect the order on risk vectors, while (P2) and (P3) require acceptable convex combinations to be achievable at bounded or interpolated adjustment cost.(P3) implies (P2).Source: §2 · pp. 13–14

Definition

  • The measure assigns each adjustment vector Y a cost π(Y) and an acceptance set AY of risk vectors acceptable for that Y; it is the infimum cost of adjustments making X acceptable.The framework allows the measure to take infinite values; additional conditions can ensure finiteness or properness.Source: §2 · pp. 13–14

Structural properties

  • Under (P1) and (P2), the systemic risk measure is monotone decreasing and quasi-convex; under (P1) and (P3), it is monotone decreasing and convex where finite.The alternative conditions yield quasi-convexity under (P1), (P2a), (P4), and (P5), and convexity under (P1), (P3a), (P4), and (P6).Source: §2 · pp. 14–16
  • The alternative conditions use combinations of adjustments: (P2a) allows some mixture weight, whereas (P3a) uses the same weight as the risk-vector mixture.With the accompanying conditions, these alternatives imply the quasi-convexity and convexity results stated above.Source: §2 · p. 16

3 Systemic Risk Measures via Aggregation and One-dimensional Acceptance Sets

The section develops systemic risk measures from aggregation functions and one-dimensional acceptance sets, with conditions for monotonicity and convexity or quasi-convexity. Its framework accommodates both adding capital before aggregation and adding it afterward, with some forms requiring additional restrictions.

3 Systemic Risk Measures via Aggregation and One-dimensional Acceptance Sets

  • The framework leaves the permitted allocation set C broadly flexible, while concave, increasing aggregation functions preserve monotonicity and convexity when lifting a one-dimensional acceptance set to multiple dimensions.The stated results generally leave C unrestricted apart from conditions required in particular propositions.Source: §3 · p. 17
  • The class in (3.2) permits either aggregating first and then adding capital or adding capital first and then aggregating, under a concave aggregation function and a monotone, convex acceptance set.These structures include Λ(X,Y)=Λ1(X)+Λ2(Y) and Λ(X,Y)=Λ1(X+Y).Source: §3 · p. 18
  • Under the stated assumptions, the framework yields quasi-convex systemic risk measures, and convexity follows under the additional convexity assumptions.The results apply across the classes studied in the section.Source: §3 · pp. 18–19
  • The truly quasi-convex class in (3.3) does not generally allow capital to be added before aggregation, because that would lose quasi-convexity.The classes in (3.6) and (3.7) consider pre-aggregation capital only with nontrivial restrictions; (3.7) uses acceptance sets Bπ(Y) for positively linear π.Source: §3 · p. 20

4 Scenario-dependent Allocations

The section studies systemic risk measures that secure a system by allocating cash across institutions, potentially depending on the realized scenario. It establishes quasi-convexity conditions and illustrates how allocation constraints and acceptance sets affect risk comparisons.

4 Scenario-dependent Allocations

  • Scenario-dependent allocations measure the minimum total cash needed to secure the system, with the admissible set C governing which distributions across institutions are allowed.Unrestricted allocations can include negative cash transfers and cross-subsidization; more restricted sets can rule these out.Source: §4 · pp. 22–23
  • The resulting risk measure is quasi-convex, and is convex when eπ is convex and θ is constant.The result applies to the stated family of allocation sets and acceptance sets.Source: §4 · p. 24
  • With C = CR, every measure of this type can be represented as a univariate quasi-convex risk measure applied to the sum of the risk factors; typical financial systems may require more restricted allocations.The unrestricted case permits unlimited cross-subsidization.Source: §4 · p. 24

4.1 Example: Worst Case Acceptance Set

  • For the worst-case acceptance set, deterministic pre-aggregation allocations require more capital than post-aggregation injection, while scenario-dependent allocations reduce capital requirements as allocation flexibility increases.Fully flexible allocations attain the post-aggregation amount; deterministic allocations also yield a unique optimal allocation, whereas constrained scenario-dependent allocations generically have multiple optima.Source: §4.1 · pp. 27–28

4.2 Example: Expected Shortfall Acceptance Set

  • With the Expected Shortfall acceptance set, deterministic and constrained scenario-dependent pre-aggregation measures match the worst-case example, but pre-aggregation capital can exceed post-aggregation capital even with flexible allocations.The comparison depends on the interaction between the aggregation function and the acceptance set.Source: §4.2 · p. 29

5 Gaussian Systems

The Gaussian-system analysis derives optimal deterministic and scenario-dependent cash allocations, then applies them to interacting network models. Scenario-dependent allocations can reduce required capital when institutions are negatively correlated, while network structure affects the marginal distributions used to assess risk.

5.1 Deterministic Cash Allocations

  • The unique optimal deterministic allocation ranks institutions by systemic riskiness, and systemic riskiness increases with an institution’s volatility.The allocation is obtained by solving a Lagrangian system; the risk measure’s solution parameter is negative.Source: §5.1 · p. 31

5.2 A Class of Scenario-Dependent Allocations

  • The Gaussian analysis permits capital allocation to depend on whether total system positions fall below or above a critical level.Allocations vary across the two states defined by whether S is at most d or greater than d.Source: §5 · p. 30, §5.2 · p. 32
  • For negatively correlated banks, scenario-dependent allocation lowers total capital requirements; for highly positively correlated banks, it does not change them.The two-bank illustrations also report lower requirements under scenario-dependent allocation as the second bank’s standard deviation increases.Source: §5.2 · pp. 33–34, Table 1 · p. 34, Table 2 · p. 35

5.3.1 Homogeneous Network

  • In homogeneous networks, increasing liquidity lowers individual position variance and, under the Section 5.1 result, systemic risk.The homogeneous model assumes common starting values, preferences, volatilities, correlations, and critical levels.Source: §5.3.1 · pp. 36–37

5.3.2 Central Clearing Network

  • For fully connected homogeneous and central-clearing networks, variances match at order one in 1/N, while their order-1/N difference determines which has smaller variance.The sign of σcρc − σρ determines which network is more stable under the comparison described.Source: §5.3.2 · pp. 37–38

5.3.3 Heterogeneous Networks

  • For heterogeneous networks, model parameters determine marginal means and variances, which can be computed numerically to obtain optimal allocations and a bank-risk ranking.The relevant parameters include lending-borrowing preferences and individual volatilities.Source: §5.3.3 · pp. 38–39

5.3.4 An Example with Random Injections

  • In a fully connected homogeneous three-bank network, symmetry makes the optimal random-allocation adjustments zero, so injections remain constant.The equal α_i values must sum to zero, yielding α_i = 0.Source: §5.3.4 · p. 39
  • In a heterogeneous three-bank example, random allocation’s capital-saving gain is more pronounced for negative correlation with common noise.The deterministic allocations do not change across the varied correlation values because they depend only on marginal distributions.Source: §5.3.4 · p. 39, Table 3 · p. 40

6 Example: Systems on a Finite Probability Space

The finite-state example restricts scenario-dependent allocations to a chosen partition of institutions into subgroups. The resulting risk requirement depends on both the allocation restriction and the dependence among institutions.

6 Example: Systems on a Finite Probability Space

  • The partition-based allocation family restricts cash distribution to within k subgroups, ranging from deterministic allocations at k = N to scenario-dependent allocation at k = 1.The risk measure is the sum of k minimal cash amounts determined today and allocated within the subgroups at time T.Source: §6 · p. 41
  • For a fixed subgroup partition, the paper states that the optimal allocation is unique and gives explicit solutions for the risk measure and allocations.The section introduces a subfamily with r singleton groups and one remaining group of size N − r.Source: §6 · pp. 41–43
  • The framework can impose lower bounds on institution-level allocations; setting every bound to zero excludes cash withdrawals and thus cross-subsidization.The paper states that the resulting optimization can be computed using Karush-Kuhn-Tucker conditions.Source: §6 · p. 43

Example 6.2

  • 79.02 is the risk measure for deterministic allocations, while the fully unconstrained scenario-dependent case attains the minimum; grouping countermonotone pairs can reduce risk substantially.Grouping X1 and X3 yields −0.56, and grouping X2 and X3 yields 4.44; grouping comonotone X1 and X2 has little effect at 74.48.Source: §6 · p. 44, Table 4 · p. 44

A.1 Gaussian Case with Random Injections

The appendix details calculations for the Gaussian random-injection optimization, differentiating across allocation cases and using the joint distribution of each position and total system position.

A.1 Gaussian Case with Random Injections

  • For α_i > 0, the appendix expands the difference between two shortfall terms and expresses the calculation using the joint distribution and density of (X_i, S).The passage specifies that (X_i, S) is bivariate Gaussian.Source: §A.1 · p. 45
  • The appendix treats α_i < 0 by analogous calculations and uses the constraint that the α_i adjustments sum to zero.It then denotes the mean and quadratic variation of X_i by μ_i and σ_i.Source: §A.1 · p. 47
  • Differentiating the Lagrangian with respect to m_i yields a condition involving the joint distribution and density of (X_i, S), provided the denominator is nonzero.The displayed derivative calculations lead into the subsequent expression for each i.Source: §A.1 · pp. 48–49

A.2 Example on a Finite Probability Space

The appendix derives optimal allocations for the finite-state example through Lagrange multipliers. It presents the proof for the fully scenario-dependent case and indicates how the other cases follow.

A.2 Example on a Finite Probability Space

  • The proof focuses on r = 0, with the general case obtainable by the same steps when the derivatives of U_i are invertible.The appendix refers readers to Pastore [2014] for further details on the general case.Source: §A.2 · p. 49
  • The appendix computes ρ using Lagrange multipliers to minimize a function, then takes derivatives with respect to y_kj and states a condition for each fixed j = 1, …, M.Source: §A.2 · p. 49
  • The appendix obtains the systemic risk measure from the optimal c and then substitutes that value to derive the optimal allocations.The resulting allocations are stated for each state j and institution k.Source: §A.2 · p. 51
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