Chapter 4: Funding Loops and the First Betti Number
A borrows from B, B borrows from C, C borrows from A. Each bilateral contract is fine. The loop is not.
The example
Three money market funds: A, B, C.
- A holds commercial paper issued by B (A has lent to B)
- B holds commercial paper issued by C
- C holds commercial paper issued by A
Each bilateral relationship is perfectly sound. A has assessed B's credit; B has assessed C's; C has assessed A's. No bilateral risk manager sees a problem.
But A's ability to repay C depends on B repaying A, which depends on C repaying B, which depends on A repaying C. The three contracts form a funding loop: a cycle of mutual dependence that has no external anchor.
If any one firm faces a liquidity shock — even a temporary one — it propagates around the loop. A cannot pay C because B has not paid A; B cannot pay A because C has not paid B. The loop amplifies rather than absorbs the shock.
\(\beta_1\): the first Betti number
The first Betti number \(\beta_1(\Gamma)\) counts the number of independent funding loops in the obligation complex. It is the dimension of \(H^1(\Gamma)\) — the first cohomology group.
- \(\beta_1 = 0\): no funding loops; the network is a tree; shocks do not amplify through cycles
- \(\beta_1 = 1\): one independent loop; one potential amplification circuit
- \(\beta_1 = k\): \(k\) independent loops; \(k\) independent contagion channels
A tree network (no loops) cannot amplify contagion — a default propagates only downstream, not in circles. A looped network can. The 2007--2008 buildup was a period of rapidly rising \(\beta_1\): repo chains, CDO/ABCP structures, and tri-party repo created an enormous number of funding loops, each an independent contagion channel.
CIP deviation as \(H^1\)
Covered interest parity (CIP) deviation is the most precisely measurable \(H^1\) signal in financial markets.
The CIA loop for USD/EUR: borrow in USD, convert to EUR at spot, invest at EUR rate, convert back at the forward rate. If CIP holds, the round trip returns exactly one dollar. CIP deviation = the holonomy of this loop: how much more or less than one dollar the round trip yields.
Before 2008, CIP deviation was < 5bp — essentially zero. Since 2008, persistent deviations of 20--100bp have been documented across all major currency pairs. This is \(H^1 \neq 0\): the CIA loop is a non-trivial cycle in the currency obligation complex. The cost of filling the 2-simplex (the arbitrage trade) is precisely the regulatory capital cost that prevents full arbitrage — the Curvature Floor Theorem.
CIP deviation is the cleanest example of \(H^1\) risk: directly observable, persistently non-zero, arising from a structural constraint rather than a model assumption.
\(H^1\) as a leading indicator
\(\beta_1(\Gamma_t)\) rises before financial crises. As institutions extend intermediation chains in benign conditions — lending to each other through longer and longer chains, creating more and more cycles — \(\beta_1\) increases. The crisis occurs when the loops can no longer be sustained: a snap deleveraging that converts a high-\(\beta_1\) network into a disconnected one (rising \(\beta_0\)).
Empirically, \(\beta_1\) leads cascade by 2--3 reporting periods. This is the \(H^1\) early-warning signal: rising loop count in low-volatility conditions is the topological signature of a system building fragility.
The formal statement
\(H^1(\Gamma) = \ker \partial_1 / \mathrm{im}\, \partial_2\). For a graph (no 2-simplices), this simplifies to \(H^1(\Gamma) = \ker \partial_1\) — the space of cycles with no boundary. \(\beta_1 = \dim H^1 = |E| - |V| + \beta_0\), where \(|E|\) is the number of edges and \(|V|\) the number of nodes. Each independent cycle is one funding loop that bilateral risk management cannot see.