Measures of Systemic Risk

• Numerical case studies show that the multivariate measures capture systemic-risk properties missed by univariate statistics and provide significantly more information than previous approaches or EARs.Source: §1 · pp. 2–3
• A cash flow or value model maps capital allocations to stochastic outcomes, while an acceptability criterion specifies which outcomes regulators consider acceptable.Systemic risk is the set of additional-capital allocations that produce acceptable outcomes.Source: §1 · p. 2
• The paper presents an algorithm for explicitly computing the set-valued measures and illustrates its implementation in numerical case studies.Source: §1 · pp. 2–3
• Unlike post-crisis bail-out measures, capital regulation adds allocations before risk aggregation and aims to prevent crises.Source: §1 · p. 2
• The framework includes bail-out measures and initial capital requirements, relaxes finite-factor and single-aggregator restrictions, and permits complex feedback mechanisms.Source: §1 · p. 3

2.1 Eligible assets, and a cash flow or value model

• A cash flow or value model is a non-decreasing random field: higher capital allocations yield higher random system values, potentially including feedback through investment payoffs or network clearing.Examples include capital-insensitive aggregation, capital-sensitive aggregation, and models of illiquid-asset sales and resulting losses.Source: §2.1 · pp. 4–7

2.2 Systemic risk measures

• Systemic risk is the set of additional capital allocations that make a model’s random output acceptable under the regulator’s acceptance criterion.The model maps capital allocations to random outcomes; the acceptance set determines which outcomes qualify as acceptable.Source: §2 · p. 4, §2.2 · p. 7
• Systemic risk measures inherit cash-invariance and monotonicity: reducing current capital by an amount requires that much more additional capital, while a better model cannot require more.Under continuity of the capital-to-output mapping, the measure also has closed values.Source: §2.2 · p. 8
• Convex acceptance criteria and concave capital-to-output mappings yield convex sets of acceptable capital allocations.This is the section’s stated condition for convexity of the systemic risk measurement.Source: §2.2 · p. 8
• D-quasi-convexity means a diversification mechanism does not increase capital requirements when its diversified response remains acceptable at the same capital allocation.A sufficient criterion is that the mechanism dominates diversification of model outputs.Source: §2.2 · p. 9
• Concave capital-sensitive aggregation gives systemic risk measures with convex values, and a convex acceptance set implies D-quasi-convexity.The analysis also gives D-quasi-convexity for capital-insensitive aggregation when the acceptance set is convex.Source: §2.2 · p. 10
• Financial networks with market clearing fit the capital-sensitive aggregation framework, yielding D-quasi-convex systemic risk when e0 is concave in its first argument and the acceptance set is convex.Source: §2.2 · p. 11
• The conditional setting of Adrian & Brunnermeier (2016) can extend beyond CoV@R, although conditional V@R does not generally yield convexity because V@R is not convex.Source: §2.2 · p. 11
• CVMs and systemic risk measures unify cases where cash-invariance and diversification apply to system inputs or outputs, including capital investments with random payoffs.These concepts do not have the same unambiguous interpretation in classical scalar monetary risk measures.Source: §2.2 · p. 11

Efficient allocation rules

• Single-valued allocation rules may assign inappropriate risk charges to individual entities, although implementing multi-valued measures can be challenging.The paper presents this attribution issue as a fundamental challenge, not a deficiency of its framework.Source: §3 · p. 12
• Capital allocations in the orthant above a minimal point are stricter requirements than the systemic risk set itself and can constrain the banking system more than necessary.The systemic risk measure characterizes allocations that lead to acceptability while retaining visibility into system interactions and externalities.Source: §3 · p. 12
• EARs select minimal acceptable capital allocations; in non-generic cases, the selected allocation may not be unique.The rule chooses minimal points of the systemic risk set, and multiple choices occur when the minimal set contains a line segment orthogonal to the price vector.Source: §3 · pp. 13–14
• Regulatory capital prices characterize EAR allocations as acceptable allocations minimizing the total price-weighted capital cost.Each price component represents the regulator’s price for one unit of a firm’s regulatory capital.Source: §3 · p. 14
• When the minimal capital set has no line segments, the corresponding price-minimizing allocation is unique.More generally, uniqueness is generic under the stated conditions when the risk set is not a half-space.Source: §3 · p. 14
• A price inversely related to firm wealth variance assigns lower regulatory prices to higher-variance firms, yielding higher minimal capital requirements for them.A nonnegative constant sets a floor on regulatory prices.Source: §3 · p. 15

A grid search algorithm

• Grid spacing sets the accuracy of the boundary approximation, as the algorithm tests whether each grid allocation produces an acceptable outcome.The test that Y_m belongs to the acceptability set is equivalent to m belonging to R(Y).Source: §4 · pp. 16, 18
• A diagonal bisection first finds neighboring grid nodes inside and outside the risk set, after which the algorithm checks adjoining edges for boundary crossings.The procedure uses the risk set’s upper-set property to identify further boundary-containing sub-boxes.Source: §4 · p. 18
• The search terminates when every point in the region is classified; the boundary lies between the resulting inner and outer surfaces.The outer approximation can be replaced by its convex hull when convexity is desired.Source: Figure 3 · p. 17, §4 · p. 18
• High-dimensional computations require substantially more time and memory, but grouping entities with equal capital requirements can reduce dimension.The grouping approach is relevant when regulatory requirements depend on firm type.Source: §4 · p. 19
• Computing an EAR can use a convex optimization formulation, while refining boundary-containing sub-boxes on finer grids permits independent parallel computations.The method can also be adapted to improve the precision of a previously computed approximation.Source: §4 · pp. 18–19

5.1 Comparative statics for aggregation functions

• Capital-insensitive requirements form a line; capital-sensitive exponential aggregation can reduce requirements, while capital-sensitive loss aggregation increases them.For exponential aggregation, the capital relief holds as long as each entity's capital is not too low.Source: Figure 4 · p. 20, Figure 5 · p. 20, §5.1 · p. 22

5.2 Comparative statics for network models

• Capital requirements are highly sensitive to inter-group interconnectedness, and the relationship across network scenarios is complex.The study presents efficient-frontier approximations and three EAR values for scenarios A1–A4.Source: Figure 6 · p. 22, §5.2 · p. 27
• Increasing connectivity among large firms raises capital requirements for both groups, regardless of small-firm capital levels.The EAR minimizing total regulatory capital can instead relieve large firms at the expense of higher requirements for small firms.Source: Figure 7 · p. 25, §5.2 · p. 27
• The effect of increasing connectivity among small firms depends on large firms' capital: requirements fall when it is high and rise when it is low.In the studied example, the total-capital-minimizing EAR favors small firms when their connectivity increases, at the expense of large firms.Source: Figure 8 · p. 26, §5.2 · p. 27
• Increasing connectivity between firm groups reduces systemic risk when there are no net intergroup payments, when large firms are net borrowers, and when small firms are net borrowers.The large-firm-borrower scenarios have the same total net amount.Source: §5.2 · p. 28
• At α = 60%, capital requirements for large and mid-sized firms are insensitive to small firms’ capital holdings, as their contour lines overlap.This result is for the chosen network model.Source: Figure 10 · p. 29, §5.2 · p. 30
• As the liquid-asset share α rises, minimal capital allocations shrink; lower liquidity also widens scenario spacing because greater illiquid-asset liquidation increases price impact.The study compares α values of 0%, 20%, 40%, 60%, 80%, and 100%, with small firms’ capital levels set to zero.Source: Figure 11 · p. 29, §5.2 · p. 30

6 Conclusion

• The framework defines systemic risk measures as the set of capital allocations that make outcomes acceptable when added to current capital.The definition uses a model of relevant stochastic outcomes and an acceptability criterion; the paper presents an algorithm for explicit computation.Source: §6 · p. 30
• EARs characterize global minimizers of capital cost and provide a simple way to apply systemic risk measures to capital regulation.The paper introduces EARs as a new concept for practical regulatory application.Source: §6 · p. 30
• The framework embeds important recent contributions to systemic risk measurement.Source: §6 · p. 30
• Numerical case studies illustrate the measures as instruments for analyzing systemic risk.Source: §6 · p. 30

A Proofs

• Convexity of the acceptance set and the random-field properties imply convexity of the risk set of acceptable additional capital.Source: A Proofs · p. 33
• Cash-invariance allows the cost-minimizing weight vector for minimal allocations to be chosen independently of current capital.Source: A Proofs · p. 33
• Under the stated conditions, a minimal element of the risk set is exactly a minimizer of a linear cost w^T m for some weight vector w.The characterization assumes a non-empty, closed, convex risk set with no nonzero recession direction in the negative orthant.Source: A Proofs · p. 33
• For the fixed-point network model, iteration establishes that p(x) is concave, and the resulting function e(x) is concave and non-decreasing.The proof obtains p(x) as the pointwise limit of concave iterates.Source: A Proofs · pp. 34–35
• Continuity of the unique fixed point follows from its closed graph and the closed-mapping property of projection over a compact space.Source: A Proofs · p. 35