Bilateral · Triangular · Systemic Risk

"Just because I don't know what the connection is doesn't mean there isn't one." — Douglas Adams, The Long Dark Tea-Time of the Soul

Financial risk has three structural levels. Existing tools address the first two. EconIAC computes all three.

Level	Name	What it is	Instruments	Who manages
\(H^0\)	Bilateral risk	Consistency of bilateral prices	Forwards, swaps	Trading desks
\(H^1\)	Triangular risk	Unhedgeable residual; convexity, basis, XVA	Options, swaptions	XVA desks
\(H^2\)	Systemic risk	Mutual inconsistency of triangular risks; cascade	CCPs, central banks	Regulators, CRO

A risk is hedgeable with bilateral instruments if and only if its \(H^1\) class is trivial. The 2008 crisis was an \(H^2\) event. No regulator was computing \(H^2\).

Start here: The Topology of Risk — a plain-language primer (13 pages, no prior mathematics required).

Why does EconIAC measure inconsistency topologically?

You have a network of banks, lenders, or trading venues. Each holds a local piece of information: a capital ratio, a funding assessment, a price. The question is not "how far are these from the truth?" The question is: "do they fit together at all?" Those are different questions, and only the second one predicts crises.

The residual paradigm and what it misses

Standard econometric and risk models work like this:

Specify a model M with parameters θ
Fit θ to minimise the sum of squared residuals: Σ (data − M(data; θ))²
Report R², standard errors, residuals as diagnostics
Call the residual "noise" — measurement error or model misspecification

This works well for isolated agents. It fails for networked systems in a specific way.

Consider a repo market under stress. MMF funds assess dealer A as solvent and roll their repo. LDI pension funds assess the same dealer A as insolvent and withdraw. Both may be individually consistent with the data available to them. A model fitted to aggregate data — total repo rolled, average haircut — may have R²=1. But the network is in an irreconcilable state: two agents acting on mutually inconsistent assessments of the same counterparty, simultaneously.

No improvement in model fit resolves this. The inconsistency is not measurement error. It is a structural property of the network state at that moment — and it is what causes the cascade.

The topological alternative: H¹ cohomology

EconIAC measures network inconsistency using H¹ cohomology of a cellular sheaf on the network graph.

The construction has three components:

Stalk: at each node (bank, dealer, venue), a vector space holding the local information — the capital ratio, the funding assessment, the asset price.

Restriction map: on each edge (bilateral exposure, lending relationship, arbitrage link), a linear map specifying how much the information at one node should agree with the information at the adjacent node, weighted by the strength of the relationship.

Coboundary operator δ₀: maps the section (the collection of all local values) to the space of edge disagreements. δ₀ applied to a globally consistent section gives zero. Applied to an inconsistent section it gives a non-zero vector of disagreements.

The H¹ signal is then:

H¹ = ‖L_F · s‖² / ‖s‖²

where L_F = δ₀ᵀδ₀ is the sheaf Laplacian and s is the section.

H¹ = 0: the local pieces fit together into a globally consistent picture. The network is in a reconcilable state.
H¹ > 0: the local pieces cannot be globally reconciled. The network is in a state of topological inconsistency. The magnitude measures how far it is from reconcilability.

Worked example: independent vs. dependent funding loops

Paper 426 measures network fragility with a related but distinct quantity: \(\beta_1\), the number of independent funding loops, defined as the first Betti number of the exposure graph, \(\beta_1 = m - n + c\) (edges minus nodes plus connected components). "Independent" is doing real work in that phrase, and it does not mean what it sounds like on first read — it is not about whether two loops look different, or share edges, or share no edges. It is about whether one loop can be written as a sum of others plus a boundary (a trivial back-and-forth contribution that adds nothing new).

Take four institutions \(A, B, C, D\) with obligations:

A → B,  B → C,  C → A,  C → D,  D → A

\(A \to B\) means A has a gross obligation to B — A owes B money. The arrow points from obligor to creditor. These are gross exposures: bilateral netting is not assumed. This is the right setting for repo and securities-lending markets, where rehypothecation means that gross positions can be many times net positions, and where a net position of zero can still sit inside a large funding loop that matters for systemic risk.

Here \(n=4\) nodes, \(m=5\) edges, \(c=1\) component, so \(\beta_1 = 5 - 4 + 1 = 2\): the network has exactly two independent funding loops, not five (the number of edges) and not one (the number of "obvious" triangles).

Three loops are visible by eye:

Loop 1 (\(A\to B\to C\to A\)): the triangle through \(B\).
Loop 2 (\(C\to D\to A\to C\), i.e. traversing \(C\to A\) backwards): the triangle through \(D\).
Loop 3 (\(A\to B\to C\to D\to A\)): the quadrilateral around the outside.

\(\beta_1=2\) says only two of these three are independent — and indeed, Loop 3 is exactly Loop 1 + Loop 2: travelling \(A\to B\to C\) (shared with Loop 1), then \(C \to D \to A\) (shared with Loop 2), is the same net obligation pattern as doing Loop 1 and Loop 2 separately and cancelling the shared \(C\to A\) / \(A\to C\) leg. Loop 3 is therefore dependent on Loops 1 and 2 — it adds no new systemic fragility beyond what Loops 1 and 2 already represent, even though it is a perfectly real, traceable cycle of obligations. A regulator who counted "how many cycles can I find by inspection" would see three and overstate the network's fragility; \(\beta_1\) correctly reports two.

This is the homological content of "financial redundancy": a loop adds systemic fragility only if it cannot be re-expressed as a combination of other loops already counted. Loops 1 and 2, by contrast, share no edges and have no such relation — both are needed to generate the full cycle space, so both count.

Direction matters only through sign, not through which way arrows point. Loop 2 above was written traversing \(C\to A\) backwards (against its stated direction, \(A\) owes \(C\)) precisely because the loop closes that way — in the signed, ℝ-coefficient chain complex this is legitimate: the edge contributes with a \(-1\) coefficient instead of \(+1\), and the cycle still closes (its boundary is still zero). A loop that runs against an edge's stated direction is not a different kind of object from one that runs with it, and counts toward \(\beta_1\) in exactly the same way — see β₁ for why edge orientation is a sign convention, not a separate cycle-counting rule.

Netting reduces \(\beta_1\); CCP novation fills triangles. Bilateral netting between \(A\) and \(C\) would remove the edge \(C \to A\) from the graph — Loop 1 (\(A \to B \to C \to A\)) collapses, the triangle disappears, and \(\beta_1\) drops from 2 to 1. This is the topological content of bilateral netting: it is an instrument that removes edges from the exposure graph, reducing the cycle rank. A CCP that novates all three legs of Loop 1 simultaneously does something stronger: it adds the triangular face as a filled-in 2-simplex, making the loop a boundary (something that bounds a filled region) rather than a free cycle. In homological terms, bilateral netting lowers \(m\) and therefore \(\beta_1\); CCP novation moves a cycle from \(H^1\) into the image of the boundary map, annihilating it as a generator of \(H^1\) entirely. Both reduce \(\beta_1\), but by different mechanisms — and only CCP novation gives the multilateral guarantee that makes the reduction robust to the default of one leg.

What makes this different from a residual

The conceptual gap is subtle but load-bearing:

	Residual	H¹ cohomology
Requires a model	Yes — residual is (data − model prediction)	No — only requires the network and a consistency relation
Node or edge property	Node property (individual deviation)	Edge property (bilateral disagreement)
Zero means	Perfect model fit	Globally reconcilable state
Non-zero means	Model misspecification or noise	Irreconcilable local assessments
Symmetric?	Yes — overfit and underfit penalised equally	No — H¹ is a topological invariant
Predictive?	Contemporaneous	Leads the cascade by 1–3 periods

The last row is the key result (Theorem 1 of Paper 335):

In any Gibbs-lifted contagion model, H¹ peaks strictly before the cascade peaks.

The intuition: the Gibbs lift creates a smooth sigmoid around the hard threshold. The bilateral inconsistency — agents beginning to disagree on the same counterparty's creditworthiness — starts accumulating as soon as the active region of the sigmoid is entered, before any individual threshold is individually breached. H¹ detects this accumulation; the cascade count only fires when the threshold is crossed.

In the CHZ fire-sale model (Paper 332) and the sovereign repo model (Paper 333), the empirical lead time is 2–3 periods.

The residual-obstruction separation

There is a stronger result (Theorem 2 of Paper 335):

A model can achieve R²=1 and H¹ ≠ 0 simultaneously.

Proof by construction: build a repo market model that correctly predicts total funding rolled (aggregate prediction = data, zero residual), but assigns the rolled funding across the dealer-lender bilateral pairs in a way that is inconsistent with the collateral coverage ratios. The residual is zero; H¹ is non-zero.

This means H¹ is not computable from a model's residuals. It measures something the standard toolkit cannot see — and something that is directly relevant to whether a cascade is imminent.

Why H¹ transfers across systems

Theorem 3 of Paper 335 (Universality) says the H¹ signal depends only on the graph topology and the restriction maps, not on the specific dynamics generating the section values. This is a structural fact about the construction, not an empirical claim about any particular pair of systems: CHZ interbank contagion (Paper 332), the sovereign repo run (Paper 333), and other section-on-a-graph data are all instances of the same H¹ machinery and inherit the same lead-time and separation guarantees from Section 2's definitions, because they share the same restriction-map structure.

We do not claim — and have not measured — that the resulting time series are quantitatively similar across systems that are not financial (the construction has been suggested as relevant to physically unrelated processes such as photosynthetic energy transfer, but no cross-correlation between that and a financial instance has been computed; treat any such claim elsewhere as a structural conjecture, not a result).

From H¹ to H²: it takes a tetrahedron, not just two triangles

H¹ measures whether one set of bilateral rates closes consistently around one triangle. It is tempting to think H² is just "two triangles disagreeing" — for instance, two dealers quoting different EUR/JPY crosses even though each dealer's own book is internally consistent. That is not quite right, and the difference matters. Two sources disagreeing about a single triangle is visible by looking at that one triangle alone (just give each edge a pair of quotes instead of one, and ask the pair to agree) — it is a richer instance of H¹, indexed by source as well as by edge, not a new, higher obstruction. No tetrahedron is needed to see it, and none is doing any work.

Genuine H² is a strictly higher-degree statement. It is not about two sources disagreeing on one triangle — it is about whether the H¹ residuals already computed on the four faces of a tetrahedron are mutually consistent with each other. Take four institutions \(A,B,C,D\), forming four face-triangles \((A,B,C)\), \((A,B,D)\), \((A,C,D)\), \((B,C,D)\) — the hollow boundary of a tetrahedron, not the solid tetrahedron itself. The distinction matters: if the solid tetrahedron were included as a data-bearing cell in its own right, the resulting shape would be a filled-in 3-ball — contractible, like a filled-in triangle is a contractible 2-disc — and its H² would vanish identically for any data, by the same mechanism that forces H¹ of a filled-in triangle to vanish. Leaving the tetrahedron hollow, so that its four faces form a closed shell topologically equivalent to a sphere \(S^2\), is what allows the four residuals on those faces to fail to close. Each face can carry its own correlation-implied H¹ residual. If all four are computed from one consistent data source, they automatically satisfy the simplicial boundary identity

c_BCD − c_ACD + c_ABD − c_ABC = 0

— this is the Pentagon identity (δ²∘δ¹ = 0) in its simplest form, and it is a tautology, with zero empirical content, for a single source. One source, however many triangles, cannot generate a non-trivial H² class.

A genuine H² obstruction needs the four face residuals to come from more than one source. Suppose three faces are priced from one consistent correlation model: \(c_{ABC}=0.012\), \(c_{ABD}=0.031\), \(c_{ACD}=0.043\). Consistency forces the fourth face to be \(c_{BCD} = c_{ACD} - c_{ABD} + c_{ABC} = 0.024\). Now suppose face \(BCD\) is instead priced by a different desk, with a different correlation assumption for that specific triple, reporting \(c_{BCD} = 0.042\) — only \(0.018\) away from the forced value, and not obviously anomalous as a number on its own. Each face individually still looks fine: no triangle's own residual is unusual, and no institution's book is internally inconsistent. But

c_BCD − c_ACD + c_ABD − c_ABC = 0.042 − 0.043 + 0.031 − 0.012 = 0.018 ≠ 0

so H² ≠ 0 on this tetrahedron. The obstruction is invisible at every triangle taken alone and becomes visible only when the four faces are checked against each other — exactly the mechanism behind Paper 397's account of correlation risk in 2008: no single desk's triangle looked wrong, but the desks' independently-priced, overlapping correlation triangles failed to close around the tetrahedron once correlations spiked. That closed-boundary failure, not any pairwise disagreement, is what aggregate or single-institution risk measures cannot see.

What H¹ requires — and what it does not

Requires: - The bilateral exposure matrix (which agent is exposed to which, at what weight) - The current section values (capital ratios, funding ratios, prices) - A consistency relation on each edge (how much adjacent nodes should agree)

Does not require: - A model of the true state - Assumptions about the distribution of errors - Parameter estimation - A prediction of what the section should be

This is what makes H¹ a genuinely different instrument from model-based risk measures. VaR, Expected Shortfall, and eigenvalue-based fragility measures (Acemoglu et al. 2015) all require a model. H¹ requires only the network and the current state.

Regulatory implications

If H¹ is a reliable 2–3 period leading indicator for cascades, and if it requires only bilateral exposure data (already reported to regulators via EMIR, FSB-LEI, and ECB repo statistics), then:

A real-time H¹ monitor is computationally feasible — it is a matrix-vector product on the bilateral exposure matrix, updated at each reporting period.
The FSB's current early-warning toolkit (based on haircut levels, concentration ratios, and VaR) could be supplemented with an H¹ signal that detects bilateral inconsistency before any individual threshold is breached.
Optimal haircut calibration can target H¹ = 0 rather than minimising systemic loss after the fact — the policy gradient ∂H¹/∂h gives the regulator the cheapest intervention to restore global reconcilability.