Systemic Risk & Stabilityadvanced

CoVaR and ΔCoVaR

CoVaR (Adrian & Brunnermeier, 2016) measures an institution’s contribution to system-wide risk. Formally, $\text{CoVaR}_q^{\,i}$ is the Value at Risk of the financial system conditional on institution $i$ being in a particular state. The headline systemic measure is ΔCoVaR: the difference between the system’s VaR when $i$ is in distress (at its own $q$ -quantile loss) and the system’s VaR when $i$ is at its median (normal) state. ΔCoVaR therefore isolates the marginal spillover that institution $i$ adds to the system. The key contrast with ordinary VaR: VaR asks "how bad can things get for firm $i$ ?"; CoVaR asks "how bad do things get for the system when firm $i$ is in trouble?"

Try it yourself

CoVaR and ΔCoVaR — a bank's footprint on system risk

Turn the camera around: not the bank's own loss, but the system's loss when this bank is in trouble. CoVaR is the system VaR conditional on the bank in distress; ΔCoVaRis that minus the system VaR when the bank is at its median state. ΔCoVaR is the bank's marginal spillover, not the same as CoVaR. A bigger, more linked bank widens the gap.

ΔCoVaR (marginal system-risk contribution)+2.07 pp

CoVaR (distress) 6.36%CoVaR (median) 4.29%ΔCoVaR +2.07% pp

ΔCoVaR = CoVaR(distress) − CoVaR(median) — the marginal gap, not the level

Linkage / dependence ρ0.60

Bank size / system weight35%

Tail level q5.0%

CoVaR — a level: the system's VaR while the bank sits in distress (its own q-quantile loss).

ΔCoVaR — a difference: CoVaR(distress) − CoVaR(median). It nets out the system's baseline and isolates the bank's marginal spillover.

Not the same. A high CoVaR can be mostly baseline system stress; ΔCoVaR is the part the bank adds.

This bank's distress lifts system VaR only from 4.29% to 6.36%, a ΔCoVaR of 2.07%. A small or weakly linked bank adds little system risk even if its own VaR is large.

ΔCoVaR (Adrian & Brunnermeier, 2016) ranks systemic importance by a firm's marginal spillover. A sibling measure, SRISK, instead asks how big a capital hole the firm would open in a crisis.

Why it matters

Ordinary VaR is a firm looking in the mirror: my own worst loss. CoVaR turns the camera around to face the whole system and asks what happens to everyone else when I am sick. A firm can have a modest own-VaR yet a large ΔCoVaR if its troubles tend to coincide with, or trigger, system-wide stress. That is exactly the firm a macroprudential regulator should watch, because the danger it poses lives in its spillover, not in its standalone loss.

Formulas

CoVaR (system VaR conditional on institution i)

\Pr\!\left(X^{\text{sys}} \le \text{CoVaR}_q^{\,\text{sys}\mid i} \;\middle|\; X^i = \text{VaR}_q^{\,i}\right) = q

CoVaR is the

q

-quantile of the system loss

X^{\text{sys}}

given that institution

i

realizes its own

q

-quantile loss (distress). The system is conditioned on the firm, not the firm on the system.

ΔCoVaR (marginal systemic contribution)

\Delta\text{CoVaR}_q^{\,i} = \text{CoVaR}_q^{\,\text{sys}\mid X^i = \text{VaR}_q^{\,i}} \;-\; \text{CoVaR}_q^{\,\text{sys}\mid X^i = \text{Median}^{\,i}}

Distress state minus median state. It captures the extra system VaR attributable to institution

i

moving from normal to distressed. Larger

\Delta\text{CoVaR}

= larger systemic footprint.

Worked examples

Scenario

Bank A and Bank B have the same standalone VaR, but A’s losses historically coincide with sharp system-wide stress while B’s do not. Which has the larger ΔCoVaR, and why does it matter?

Solution

Bank A. Because the system’s VaR rises much more when A is in distress than when A is normal, A’s ΔCoVaR is large even though its own VaR equals B’s. ΔCoVaR reveals that A is more systemically important: a regulator concerned with stability should impose tighter requirements on A, a distinction that firm-level VaR alone would miss.

Scenario

Why is ΔCoVaR, not CoVaR or VaR, the quantity Adrian and Brunnermeier emphasize for ranking systemic importance?

Solution

CoVaR conditional on distress still mixes in how stressed the whole system already is. Subtracting the median-state CoVaR nets that out and isolates the marginal contribution of the specific institution moving from normal to distressed. ΔCoVaR is thus the clean measure of how much extra system risk a given firm adds, which is what macroprudential ranking needs.

Common mistakes

✗CoVaR is the same as VaR, just larger. They condition in opposite directions. VaR is the firm’s own loss quantile; CoVaR is the system’s loss quantile conditional on the firm being in distress.
✗CoVaR and ΔCoVaR are interchangeable. CoVaR is a conditional level; ΔCoVaR is the difference between the distress-state and median-state CoVaR, isolating the firm’s marginal systemic contribution.
✗A firm with low standalone VaR cannot be systemically important. A firm can have a small own-VaR yet a large ΔCoVaR if its distress coincides with or amplifies system-wide stress.

Revision bullets

•CoVaR: VaR of the system conditional on institution i’s state
•Conditions the system on the firm, the reverse of ordinary VaR
•ΔCoVaR = CoVaR(i in distress) minus CoVaR(i at median)
•ΔCoVaR isolates a firm’s marginal contribution to system risk
•A low own-VaR firm can still have a large ΔCoVaR

Quick check

CoVaR for institution $i$ is defined as

ΔCoVaR is computed as the difference between the system’s VaR when institution $i$ is

Connected topics

Systemic Risk Systemic vs Systematic MES / SRISK Macroprudential

Sources

Adrian & Brunnermeier (2016)
Adrian, T., and Brunnermeier, M. K. "CoVaR." American Economic Review 106 (7), 2016, 1705-1741.
Original definition of CoVaR and ΔCoVaR as measures of a firm’s contribution to systemic risk.
Nguyen, V. P. (Thesis, Ch. 3)
Nguyen, V. P. Bank Efficiency and Systemic Risk. Doctoral thesis, Chapter 3. (Author’s own research; estimates CoVaR/ΔCoVaR alongside MES and SRISK.)
Dr. Nguyen’s thesis applies CoVaR/ΔCoVaR with MES and SRISK to study the link between bank efficiency and systemic risk.

How to cite this page

Dr. Phil's Quant Lab. (2026). CoVaR and ΔCoVaR. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-covar

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