CoVaR and Marginal Expected Shortfall

A hospital ward can ask two very different questions about one patient. "How sick could this patient get on their own worst day?" is an ordinary, individual question. "How much sicker does the whole ward get on the days this patient is contagious?" is a different question entirely — it's about spillover, not just severity. Traditional VaR only ever asks the first question about a bank or fund: how bad can this book's own losses get. CoVaR and Marginal Expected Shortfall (MES) ask the second kind of question, in both directions — one asks how much worse the system gets because of the institution, the other asks how much worse the institution gets because of the system.

Two questions, two directions

CoVaR conditions the system's risk on the institution being in distress:

$$P\big(R_{\text{system}} \le \text{CoVaR}_{q \mid i} \;\big|\; R_i = \text{VaR}_{i,q}\big) = q .$$

In words: $\text{VaR}_{i,q}$ is institution $i$'s own ordinary VaR at confidence level $q$ — its bad-day threshold. $\text{CoVaR}_{q \mid i}$ is the system's VaR, but computed conditional on institution $i$ already sitting exactly at that bad-day threshold. The institution's true systemic footprint, $\Delta\text{CoVaR}_i$, is this conditional system VaR minus the system's VaR when the institution is in its normal, median state — the extra system-wide risk that shows up specifically because this one institution is in distress.

Marginal Expected Shortfall runs the conditioning the other way:

$$\text{MES}_i = \mathbb{E}\big[R_i \mid R_{\text{system}} \le \text{VaR}_{\text{system},q}\big] .$$

In words: take every day the system was in its own worst-$q\%$ tail, and average institution $i$'s return specifically on those days. A high (very negative) MES means the institution tends to fall hardest exactly when everything else is already falling — it doesn't diversify away in a systemic event, it amplifies it.

Worked example 1 — computing MES from tail days

A market has 100 trading days of data; the worst 5 days for the overall system (the 5% tail) had system returns of $-6\%, -5.5\%, -5\%, -4.5\%, -4\%$. A particular bank's stock returned $-9\%, -7\%, -8\%, -6\%, -5\%$ on those exact same five days. Its MES is the simple average of its own returns on the system's worst days: $(-9 - 7 - 8 - 6 - 5)/5 = -35/5 = -7\%$. Compare that to the bank's own unconditional average daily return of, say, $0.05\%$ — on the days the system is in crisis, this bank loses on average 140 times worse than its typical day, a sign it's a systemic amplifier rather than a diversifier.

Worked example 2 — reading a ΔCoVaR number

Suppose a regulator estimates that when a large dealer bank is at its own 1% VaR (a genuinely bad day for that bank alone), the system's 1% VaR is $-8\%$; but when the same bank is at its median, normal-day return, the system's 1% VaR is only $-3\%$. Then $\Delta\text{CoVaR} = -8\% - (-3\%) = -5\%$: this one bank's distress is associated with an extra 5 percentage points of system-wide tail risk, over and above what the system experiences on an ordinary day for that bank. A second, smaller bank might show a $\Delta\text{CoVaR}$ of only $-0.5\%$ — individually just as capable of losing money on its own worst day, but contributing far less to system-wide risk because its distress doesn't spill over.

Slide the mean down to simulate a system in distress. Ordinary VaR asks about the left tail of one institution's own curve; MES asks what that institution's curve looks like specifically during the days sitting in the system's left tail — a much narrower, more dangerous slice of days than the institution's own history.

What this means in practice

Regulators use these measures to identify systemically important institutions — a bank can have a modest standalone VaR yet a large $\Delta\text{CoVaR}$ or MES if its distress reliably coincides with, or transmits into, everyone else's. That's the basis for capital surcharges on globally systemic banks: not "how risky is this firm alone" but "how much of the system's risk runs through this firm."