4.13.3 Systemic Risk Measures

Basic Prerequisites: VaR & Expected Shortfall and 02 · Contagion & Networks.

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1. Intuition & Practical Objective

Network contagion (§02) tells you that systems can collapse; systemic-risk measures try to tell you which firm is most dangerous, so a supervisor can charge capital or demand resolution. This page covers the three workhorse measures. The objective: keep their conditioning direction straight, do the mathematics correctly, and compute them on a tiny system where you can see what each one does.

The three questions a supervisor might ask, and which measure answers it:

Question	Measure	It conditions on…
”How much does firm i lose when the system is in trouble?”	MES - marginal expected shortfall	the system’s tail
”How much does the system lose when firm i is in trouble?”	CoVaR - conditional VaR	the firm’s tail
”How much capital will firm i need in a crisis?”	SRISK - capital shortfall	an explicit crisis scenario

MES and CoVaR are opposite directions of the same conditioning. Getting them backwards is the single most common conceptual error.

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2. Mathematical Ground Truth & Derivations

Let firm $i$ return be $r_i$, the system (market) return be $R$, and let $\alpha$ be a small tail probability (e.g. $5\%$).

MES (Acharya–Pedersen–Philippon–Richardson). In expected-shortfall form, the system’s ES is a value-weighted sum of member MESes:

$${\text{ES}}_{\alpha }(R)=\sum _i{w}_i\mathbb{E}[r_i\mid R\le {\text{VaR}}_{\alpha }(R)]\equiv \sum _i{w}_i{\text{MES}}_{i,\alpha }.$$

Intuitively each firm contributes to system tail risk in proportion to its own expected return inside the system’s tail. A firm that is spread-hurting-but-small can have large MES even if its weight $w_i$ is modest.

CoVaR (Adrian–Brunnermeier). CoVaR is the VaR of the system conditional on firm $i$ being at its own VaR:

$$\mathrm{Pr}(X^{\text{system}}\le {\text{CoVaR}}_i^{\alpha }|X^i={\text{VaR}}_i^{\alpha })=\alpha ,$$

and the ΔCoVaR is the contribution - how much worse the system’s tail is when firm $i$ is distressed than in its median state:

$$\Delta {\text{CoVaR}}_i={\text{CoVaR}}_i^{\alpha }-{\text{CoVaR}}_i^{50\%},$$

(the textbook baseline is the firm’s median state ${\text{VaR}}_i^{50\%}$; the common practical simplification - used by this page’s code and the hub - takes the unconditional system VaR as the baseline, which is the figure quoted below). Note it is a difference of conditional quantiles, not a sensitivity/derivative - ΔCoVaR is not a calculus gradient, a point often muddled in practice.

SRISK (Brownlees–Engle). A crisis scenario (cumulative market loss over some horizon ≥ threshold). Each firm’s capital shortfall is

$${\text{SRISK}}_i=\mathbb{E}[k{A}_i-E_i\mid \text{crisis}{]}_{+},$$

where $A_i$ = assets, $E_i$ = equity, $k$ = required capital ratio (≈8% post-crisis). SRISK is the amount of fresh capital firm $i$ would need to keep ratio $k$ in the crisis - an economic-resources measure rather than a tail-quantile one, and the one regulators most directly convert into “systemically important” designations.

The three agree on direction, not on magnitude - a bank can be top-MES and bottom-SRISK (e.g. a high-marginal-risk but deeply-capitalised firm). Use all three, never one alone.

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3. Computational Implementation - CoVaR & MES on a 2-bank system

We build a two-bank system from a common factor $F\sim N(0,1)$ plus idiosyncratic noise, so banks are correlated at $\rho ={\sigma }_F^2/({\sigma }_F^2+{\sigma }_{\text{id}}^2)=0.5$. Then we estimate, by direct simulation, the tail quantities. Stdlib only (Box–Muller).

Read the output. In the system’s worst 5% ($\le -2.0138$) bank 1 loses on average -2.52 - that’s its MES; it is exposed to the system tail. When bank 1 itself is in its worst 5%, the system’s 5% quantile drops from -2.01 to -3.42 - that’s CoVaR, the damage bank 1 does to the system; ΔCoVaR = -1.40 is bank 1’s marginal contribution. Same data, two opposite questions: MES says “how much this bank suffers in a crash,” CoVaR says “how much the crash suffers because this bank exists.” Both are larger in magnitude than the unconditional VaR - the connected bank both catches and spreads tail risk.

(SRISK is scenario-based rather than quantile-based; it is not computed here because it needs a crisis-horizon capital projection - see §2 for the formula and §5 for its literature.)

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4. Failure Modes & First-Principles Breakdowns

1. Direction confusion. Swapping the conditioning (CoVaR vs MES) answers the wrong policy question and can rank banks oppositely. Always state what is conditioned on.
2. CoVaR ≠ VaR of the stressed firm. CoVaR is a system quantile conditioned on the firm; it is not a firm-level number rescaled. (Here it is -3.42, more negative than firm VaR.)
3. Tail estimation is the weak link. All three need reliable extreme-tail inputs (high $\alpha$, crisis scenarios), which historical data under-samples - the “unmeasurable tail dependence” failure of §05 is baked in here.
4. Procyclical inputs. If the measure uses current VaR/vol, it rises exactly when the supervisor must force deleveraging (see 05 · Failure Modes). Countercyclical buffers are the patch (see 06 · Advanced Extensions).

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5. References

• Adrian & Brunnermeier, CoVaR, American Economic Review 106(7):1705–1741 (2016)
• Acharya, Pedersen, Philippon & Richardson, Measuring Systemic Risk, Review of Financial Studies 30(1):2–47 (2017)
• Brownlees & Engle, SRISK: A Conditional Capital Shortfall Measure of Systemic Risk, RFS 30(1):49–79 (2017)
• McNeil, Frey & Embrechts, Quantitative Risk Management (2015)

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6. Connected Graph Bridges

• Back: 02 · Contagion & Networks · Index Hub
• Forward: 04 · Aggregating Risk Types · 05 · Failure Modes
• Base: Coherent Risk Measures · Expected Shortfall