4.13.1 Systemic Risk & Aggregation from Zero

Basic Prerequisites: VaR & Expected Shortfall.

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1. Intuition & Practical Objective

This page builds the why of systemic risk with no prior knowledge of systemic stuff needed. The objective is one idea: a system’s risk is an emergent, nonlinear property of how its members are connected - it cannot be read off from any member’s individual risk, and small shocks can cross sharp thresholds into catastrophes that no individual model foresaw.

Start with the dumbest question: why isn’t the financial system just a sum of banks? A system of 100 banks, each with the same standalone risk number, can be either extremely robust or extremely fragile - the difference lives entirely in who owes whom. Two otherwise identical systems:

• A star (one hub): every bank lends to a central hub. If the hub fails, everyone loses.
• A ring: each bank lends to its neighbour. A loss hits one, passes to the next, and can circulate.

The individual banks look identical. The systems behave completely differently. That is the first “aha”: default contagion is a network phenomenon, not a firm phenomenon.

Three steps:

1. Interconnectedness is not always bad - it is a phase transition. In a sparse network a failure concentrates and wipes out several neighbours; in a very dense network each exposure is small and the loss is absorbed. The danger zone is the middle - this is the “robust-yet-fragile” result (Allen–Gale 2000; Gai–Kapadia 2010) that the §3 experiment reproduces.
2. Tail correlation is what you can’t measure. Losses from different banks / different risk types are only strongly correlated in the extreme tail - exactly where you have almost no data. The whole aggregation problem is: the quantity that matters most is the one you cannot estimate.
3. Per-firm safety can create system danger. If every bank cuts leverage when its VaR rises, they all sell the same assets at the same time - the individually rational rule is collectively destabilising (the procyclicality loop of §05).

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2. Mathematical Ground Truth & Derivations

The threshold model of contagion. Let $N$ banks each hold capital $E$ and interbank claims. Suppose bank $i$ has lent $x_i>0$ to the next bank in a ring. Bank $i$ defaults when cumulative losses $\ge E$. When bank $j$ defaults, its creditor (say $k$) loses its claim $x_k$ on $j$:

$${\text{default}}_j\Rightarrow \text{loss to creditor }k=x_k\ge E\Rightarrow {\text{default}}_k,$$

which can cascade around the whole ring. The elasticity of the cascade to a single external shock $s$ hitting bank 0 is the number of banks that default. For a ring with equal $E$ and equal exposure $x=8>E=5$:

• $s\le E=5$: absorbed - 0 defaults (the shock is contained by the bank’s own capital).
• $s>E$: bank 0 fails, its creditor loses $8=E+3$, fails, and so on around the ring - all 5 default.

So the same system is safe for shocks below $E$ and totally destroyed for shocks above it. That discontinuity - a first-order phase transition - is the mathematical signature of systemic risk, and it is invisible to any single-bank risk model.

The aggregation identity that isn’t. For a portfolio of two loss streams $X_1,X_2$ with any dependence, expected shortfall is subadditive:

$${\text{ES}}_{\alpha }(X_1+X_2)\le {\text{ES}}_{\alpha }(X_1)+{\text{ES}}_{\alpha }(X_2).$$

So the “naive sum” (adding each marginal ES) is an upper bound you can never actually hit unless the two streams are perfectly comonotonic (one tail = the other tail, $\rho =1$). The lesson in §04: $63.97$ (sum) vs $55.65$ (corr $0.5$) vs $45.85$ (independent). Any aggregation method that doesn’t account for dependence is either double-counting (sum) or, worse, hiding joint tail risk (treating them as independent when they are tail-linked).

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3. Computational Implementation - the phase transition in a ring

The cleanest way to see contagion is the deterministic ring cascade above. Stdlib only.

Read the output. Shocks of 1 and 4 are absorbed by bank 0’s €5 capital. A shock of 6 (just over one bank’s equity) detonates the entire ring: 0 loses 1 too much, fails; its creditor 4 loses the €8 claim, now €-3, fails; creditor 3 loses €8, fails; and so on - order [0,4,3,2,1]. The same €12 shock produces the same full collapse, not more. The system’s fragility is a threshold, not a scale. Adding more nominal capacity to every bank (raising $E$) actually raises the cascade threshold but changes nothing about the nonlinearity once crossed - the classic argument against “just make everyone bigger” as a systemic cure.

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4. Failure Modes & First-Principles Breakdowns

1. The “sum of banks” fallacy. Computing a VaR for each bank and adding them is the single most intuitive and most wrong approach; because marginal tails are subadditive, the sum overstates, while any independence assumption hides joint risk. There is no correct additive decomposition of system risk.
2. The continuity delusion. Real contagion is a threshold/cascade process (here: jump from 0 to 5 defaults at $s=6$). Modelling it with smooth correlations or a single volatility misses the discontinuity - the failure mode where a “small, diversified shock” is fine for 50 periods and then destroys the ring in period 51.
3. The completeness trap. These cascade counts assume you know the full exposure matrix. In reality interbank exposures are only partially observable (bilateral OTC, netting, off-balance-sheet), so even the graph is estimated with error - see 05 · Failure Modes.

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5. References

• Allen & Gale, Financial Contagion, Journal of Political Economy 108(1):1–33 (2000)
• Gai & Kapadia, Contagion in Financial Networks, Proc. R. Soc. A 466:2401–2423 (2010)
• Haldane, Rethinking the Financial Network (2009, Bank of England)

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6. Connected Graph Bridges

• Base: VaR & Expected Shortfall · Margin & Funding Spirals (the micro-mechanism of contagion)
• Continue: 02 · Contagion & Networks · Index Hub