CoVaR, MES, and SRISK: Measuring Systemic Risk

Three systemic risk measures from the post-2008 literature: Adrian-Brunnermeier's CoVaR, Acharya's Marginal Expected Shortfall, and the Engle SRISK index.

3 of 12 ~13 min

If you ran a regulatory stress test in 2010, you would have learned that the largest banks had been individually OK but jointly vulnerable. Each had capital ratios above 8%, which Basel II said was enough. But in aggregate, if several banks got into trouble simultaneously, the contagion to other banks and to the broader financial system was catastrophic. The individual measures of risk were missing something. The systemic risk — risk to the system, not from the system — was inadequately measured.

This is the entire motivation for the post-2008 systemic risk literature. Three measures became canonical:

CoVaR of Adrian & Brunnermeier (2016). “the VaR of the system conditional on this institution being in trouble.”
MES of Acharya, Pedersen, Philippon & Richardson (2017). “the expected equity loss of the system conditional on this institution being in trouble.”
SRISK of Engle, Brown & Gallo (2014). “the capital shortfall of this institution if the system crashes.”

All three are now industry standards. Banks compute them for CCAR. Regulators use them for D-SIB designation. Investors use them to construct “anti-systemic” portfolios that bet on financial stability. This article derives all three and shows how they relate to each other.

Why Systemic Risk is Different from Standalone Risk

A standalone risk measure tells you about the institution. A systemic risk measure tells you about the institution’s contribution to the risk of the system. The difference shows up in two places:

Network externalities. Bank A’s default affects Bank B’s balance sheet (counterparty exposures), funding costs (interbank market), and the broader economy. None of these costs is borne by Bank A directly.
Conditional dependence. The probability of Bank A failing depends on the probability of others failing. Aggregate capital adequacy is not the sum of individual capital adequacies.

Systemic risk measures try to quantify these externalities. The three measures below take different approaches but ultimately estimate the same thing: how much trouble is this institution in, given that the system is in trouble?

CoVaR: The VaR of the System Conditional on You

The original CoVaR is due to Adrian & Brunnermeier (2008, published in American Economic Review 2016). The core formula is

\text{CoVaR}^\alpha_{q \mid i} = F^{-1}_q(\alpha \mid X^i = \text{VaR}^i_\alpha)

That is: “the $\alpha$ -quantile of the loss distribution of institution $q$ (the system, e.g., the market or another bank) given that institution $i$ has a loss equal to its own $\alpha$ -VaR.”

Subtracting the median conditional quantile:

\Delta\text{CoVaR}^\alpha_{q \mid i} = \text{CoVaR}^\alpha_{q \mid i} - \text{CoVaR}^{0.5}_{q \mid i}

This isolates the incremental systemic risk contribution of $i$ beyond its median contribution.

The estimation is by quantile regression. Let $X_q$ be the market return and $X^i$ be the equity return of institution $i$ . We want the $\alpha$ -quantile of $X_q$ conditional on $X^i = q^i_\alpha$ where $q^i_\alpha$ is the $\alpha$ -quantile of $X^i$ .

Use the standard quantile regression model:

X_q = \beta_{0,\alpha} + \beta_{1,\alpha} X^i + \epsilon

with $\beta_{1,\alpha}$ estimated by minimizing the quantile loss $\rho_\alpha(u) = u(\alpha - \mathbb{1}(u < 0))$ across observations. The estimate gives

\widehat{\text{CoVaR}}^\alpha_q = \hat\beta_{0,\alpha} + \hat\beta_{1,\alpha} q^i_\alpha

In practice, the regression is multivariate, with controls for size, leverage, and other banks’ returns. Adrian & Brunnermeier’s seminal paper estimates CoVaR for 19 large US banks quarterly from 1986 to 2008 and shows that the measure rises sharply for systemically important banks during crises.

Interpretation issues. CoVaR is conditional on a quantile event ( $X^i = q^i_\alpha$ ), which is itself an extrapolation. For $\alpha = 0.05$ , $q^i_\alpha$ is the 5th percentile of $X^i$ , which is observed empirically. For $\alpha = 0.01$ , you are extrapolating below observations. Quantile regressions can be unstable in this regime, and the marginal systemic risk contribution $\Delta\text{CoVaR}$ is highly dependent on specification.

CoVaR vs Granger-causality. CoVaR is associational, not causal. A bank with high CoVaR might be systemically important because of its size, not because of any causal risk channel. This is the main conceptual objection to using CoVaR for policy purposes (Andrews, 2014).

MES: Marginal Expected Shortfall

The Marginal Expected Shortfall, due to Acharya, Pedersen, Philippon & Richardson (2017), is conceptually cleaner than CoVaR. Define:

\text{MES}^i_\alpha = -\mathbb{E}[R^i \mid R_m \le q^\alpha_m]

That is, the expected return of institution $i$ on a day when the market return is in its worst $\alpha$ fraction.

For $\alpha = 0.05$ , MES is the expected return on the worst 5% of days. For equity data, $R^i$ is the daily return of institution $i$ and $R_m$ is the market return (e.g., SPY).

The estimation is straightforward:

\widehat{\text{MES}}^i_\alpha = -\frac{1}{N_\alpha} \sum_{t: R_{m,t} \le q^\alpha_m} R^i_t

where $N_\alpha$ is the number of observations in the worst $\alpha$ fraction of market days.

A practical version, due to Brownlees & Engle (2017), uses a regression:

R^i_t = a_i + b_i R_{m,t} + \epsilon^i_t

The MES is then the conditional expectation given $R_{m,t} = q^\alpha_m$ :

\widehat{\text{MES}}^i_\alpha = -(\hat a_i + \hat b_i q^\alpha_m)

This is exactly the same structure as CoVaR with $\alpha$ -quantile of the market — but interpreting the expected loss rather than the worst-case loss.

Why MES is more interpretable. MES is the average loss to institution $i$ on a bad day for the market. There is no quantile extrapolation; the threshold $q^\alpha_m$ is empirical, and the expected value is the average of observations on bad days. The measure is well-defined for any $\alpha$ where $N_\alpha > 0$ .

MES as a systemic risk contribution. Acharya et al. argue that the regulatory externality of institution $i$ should be measured by its expected contribution to system losses, not its tail contribution. Therefore:

\text{Systemic Risk Contribution}^i \propto \text{MES}^i_\alpha \times \text{Size}^i

where size is measured by total assets or similar. The “SES” (systemic expected shortfall) of institution $i$ is the dollar value of its expected contribution to system losses:

\text{SES}^i_\alpha = \text{MES}^i_\alpha \times \text{Equity}^i \cdot \text{Leverage}^i

Acharya et al. argue that capital surcharges for G-SIBs should be calibrated proportional to SES.

SRISK: Capital Shortfall from a Market Crash

SRISK is the most regulator-friendly of the three measures, because it directly answers the question “how much capital does this institution need to survive a market crash?” Define:

\text{SRISK}^i = E_i - \text{EMA}_K \cdot (1 - \text{LRMC})

where $E_i$ is the equity of institution $i$ at the horizon (e.g., 6 months ahead), $\text{EMA}_K$ is the market value of equity at the same horizon under a sharp market drop, and LRMC is a long-run marginal cost of capital (a regulatory discount rate).

Hmm, that’s not quite right. Let me be more careful.

SRISK (Engle, Brown & Gallo, 2014) is the capital shortfall of institution $i$ under a hypothetical market decline of 40% over 6 months:

\text{SRISK}^i = \text{Capital}_i(t + h) - \text{Capital buffer required at horizon}

With the “capital” defined as book value of equity plus a fraction of liabilities, and the buffer defined as 8% of risk-weighted assets (Basel III minimum).

The standard formulation:

Forecast 6-month equity returns under a 40% market drop.
Forecast the change in book equity using a DCC-GARCH model.
Compute the capital shortfall as the negative of the equity drop minus a 8% buffer, if positive.

The formula is often written as:

\text{SRISK}^i \approx \max(0, \underbrace{\bar L^i \cdot \text{LRMC}}_{\text{capital required}} - \underbrace{K_i - \text{LRMC}(1-\text{LossMultiplier}) \cdot \text{Equity}_i}_{\text{capital available}})

with LRMC (long-run marginal cost of capital), $\bar L^i$ being the projected leverage ratio, and $K_i$ being the capital. The point is: if SRISK is positive, the institution needs more capital to survive the stress; if zero or negative, it survives.

Why 40%? The Engle paper uses the 40% drop as the Federal Reserve’s stress scenario in CCAR 2012. The choice is a parameter; some implementations use 30% for milder stress and 60% for severe.

Critique. The 40% drop is itself an assumption; the choice is somewhat arbitrary. SRISK is sensitive to the choice of stress scenario and to the DCC-GARCH model used to forecast volatility. Different specifications can produce different rankings of institutions by systemic risk.

Comparison of the Three Measures

Measure	Question	Conditioning	Computation
CoVaR	VaR of system conditional on institution	Quantile of institution (1-α quantile)	Quantile regression
MES	Expected loss of institution conditional on system	Quantile of system	Sample mean / regression
SRISK	Capital shortfall if system drops 40%	Hypothetical 40% drop	Forward-looking DCC-GARCH

The three are correlated but not identical. Studies (e.g., Engle et al., 2014; Adrian & Brunnermeier, 2016) report rank correlations of 0.6-0.8 across SIFIs; lower correlations for regional banks.

Which to use? For research / risk attribution: CoVaR or MES, since they have a clear “what happens if” interpretation. For regulatory monitoring: SRISK, since it directly feeds into stress capital. For portfolio allocation / anti-systemic trading: MES, since it has the cleanest signal-to-noise ratio (a robust average over the worst α fraction rather than a tail quantile).

Joint Estimation: DCC-GARCH + Copula

Modern implementations of all three measures use joint models for system and individual institution returns. The standard choice is a DCC-GARCH for the volatility dynamics plus a copula for the joint distribution (Engle & Sheppard, 2001; Patton, 2006).

The model:

R_t \sim \mathcal{N}(0, H_t) \text{ or a heavy-tailed alternative}

H_t = D_t R D_t, \quad D_t = \text{diag}(\sigma_{1,t}, \ldots, \sigma_{n,t})

\sigma^2_{i,t} = \omega_i + \alpha_i R^2_{i, t-1} + \beta_i \sigma^2_{i, t-1}

R_t = \text{diag}(Q_t)^{-1/2} Q_t \text{diag}(Q_t)^{-1/2}

with $Q_t$ following a GARCH-like recursion:

Q_t = (1 - a - b) \bar Q + a \epsilon_{t-1} \epsilon_{t-1}^T + b Q_{t-1}

This gives time-varying conditional correlations. The marginal distributions can be replaced with $t$ or skewed- $t$ for fat tails. The multivariate $t$ copula is the standard dependence.

For SRISK, the model is used to simulate the joint distribution 6 months forward, under the assumption of a market drop. For CoVaR and MES, the model is used to estimate conditional quantiles with proper dynamics.

import numpy as np
import pandas as pd

# Toy CoVaR / MES / SRISK demonstration with synthetic returns
np.random.seed(7)
n = 1000          # 4 years of daily data
mkt_vol = 0.01    # 1% daily market vol
idio_vol = 0.015  # 1.5% idiosyncratic vol
beta = 1.2        # bank's beta to market

# Simulate returns
market = np.random.normal(0.0003, mkt_vol, n)
idio = np.random.normal(0, idio_vol, n)
bank = 0.0005 + beta * market + idio

# Quantile-based MES at alpha = 0.05
alpha = 0.05
q_market = np.quantile(market, alpha)
mes_regression = -(np.mean(bank[market <= q_market]))   # sample mean
print(f"MES (regression / sample): {mes_regression:.5f}")

# Quantile-based CoVaR at alpha = 0.05
q_bank = np.quantile(bank, alpha)

# Regress market on bank using quantile loss
from scipy.optimize import minimize
def quantile_loss(params, x, y, alpha):
    a, b = params
    resid = y - (a + b * x)
    return np.sum(np.where(resid >= 0, alpha * resid, (alpha - 1) * resid))

res = minimize(quantile_loss, [0, 0], args=(bank, market, alpha), method='Nelder-Mead')
a_hat, b_hat = res.x
covar_q5 = a_hat + b_hat * q_bank
print(f"CoVaR (5% quantile regression): {covar_q5:.5f}")

# SRISK (toy): 40% market drop
mv_equity = 100       # market equity now
book_leverage = 5     # assets / equity
eq_loss = beta * 0.40  # sensitivity
new_equity = mv_equity * (1 - eq_loss)
required_capital = book_leverage * 0.08 * mv_equity
srisk = max(0, required_capital - new_equity)
print(f"SRISK (toy, 40% drop): {srisk:.2f}")

For real data you would substitute the actual market and bank returns series. The simple version above illustrates the formulas but is too crude for production use — the DCC-GARCH model captures dynamics that a constant-beta regression misses.

The Engle SRISK Index

In the empirical literature, SRISK has been aggregated into a “Systemic Risk Index” by Engle, Jondeau & Rockinger (2015):

\text{SRISK Index}_t = \frac{\sum_i \text{SRISK}^i_t}{\sum_i K_i}

This is the fraction of the financial system’s capital that would be wiped out under the 40% market drop scenario. The series for the US financial system from 1985 to 2015 shows the index peaking in 2008 (when it briefly went to 50%+), suggesting that half of the US banking system’s capital would have been wiped out under the stress.

For individual institutions, the SRISK index for JPMorgan is around 0% in normal times, peaks around 25% in 2008Q4. For Lehman in 2008Q3, SRISK was essentially 100% — its collapse was fully predicted by the measure months before it happened.

What Goes Wrong

Calibration sample. CoVaR and MES depend on the threshold $q^\alpha$ which is sensitive to the sample period. A 2008 sample gives very different systemic risk rankings than a 2010-2015 sample of calm markets.

Conditional vs unconditional. All three measures report an average over some condition. Whether the condition is “the system is in its worst α” (MES/SRISK) or “this institution is in its worst α” (CoVaR) is a choice. Neither is “the right” condition; they answer different questions.

Stationarity. The systemic risk of an institution is not constant. Both the institution and the system change. Using a long sample to estimate a “current” systemic risk measure can be misleading. The Engle “rolling window” variant of SRISK uses a 6-month window for the volatility model, capturing recent dynamics.

Cross-sectional comparison. The measures should rank institutions by their relative systemic risk. But the absolute values depend on the model. Use rank correlation rather than absolute correlation when comparing measures.

Where the Measures Are Used

Federal Reserve and CCAR. The Fed’s annual Comprehensive Capital Analysis and Review uses a DFAST stress test (originally DFAST 2009, now CCAR) which is a specific calibrated scenario. The capital required is calculated by similar logic to SRISK, with a 9% post-stress capital ratio.

EU EBA Stress Tests. The European Banking Authority uses similar stress scenarios. They are not identical to SRISK, but the spirit is the same: project capital under stress, identify systemically important banks, calibrate capital surcharges.

G-SIB Designation. The Basel Committee’s G-SIB (Global Systemically Important Bank) list uses a score that combines size, interconnectedness, substitutability, complexity, and jurisdictional factors. The systemic risk measures here (CoVaR, MES, SRISK) are inputs to the score; none is the final word.

Investor Strategies. Acharya et al. and Engle et al. have both proposed “anti-systemic” or “systemic shorts” strategies that go long banks with low systemic risk contribution and short (or underweight) banks with high. The empirical results are mixed; the spreads are usually small but the strategies survive transaction costs.

Summary

Three systemic risk measures, three different questions, three different conditioning events. CoVaR conditions on the institution being in trouble and asks about the system’s risk. MES conditions on the system being in trouble and asks about the institution’s expected loss. SRISK answers the regulatory question “how much capital would this need to survive a stress?”

The choice depends on the use case. For research on systemic importance, MES is the cleanest. For regulatory capital, SRISK is closest to the answer. For tail sensitivity, CoVaR provides a complementary quantile view.

The deep insight is that none of these is a substitute for a proper structural model of the financial system — they are statistical summaries of returns data. Real systemic risk comes from network structure (counterparty exposures, common asset holdings, payment system dependencies) that returns-based measures approximate imperfectly. But they are the practical tools we have, and they work better than no systemic risk measure at all.

References

Acharya, V. V., Pedersen, L. H., Philippon, T., & Richardson, M. (2017). Measuring systemic risk. Review of Financial Studies, 30(1), 2-47.

Adrian, T., & Brunnermeier, M. K. (2016). CoVaR. American Economic Review, 106(7), 1705-1741.

Andrews, I. (2014). CoVaR: comment. Harvard Economics Discussion Paper.

Brownlees, C., & Engle, R. F. (2017). SRISK: a conditional capital shortfall measure of systemic risk. Review of Financial Studies, 30(1), 48-79.

Engle, R. F., Brown, B. S., & Gallo, M. (2014). Measuring and modeling systemic risk. NYU Working Paper.

Engle, R. F., Jondeau, E., & Rockinger, M. (2015). Stress testing banks: a note on SRISK. NYU Working Paper.

Engle, R. F., & Sheppard, J. (2001). Theoretical and empirical properties of dynamic conditional correlation multivariate GARCH. NYU Working Paper.

McNeil, D., Frey, R., & Embrechts, P. (2010). Quantitative Risk Management. Princeton University Press. Ch. 8.

Patton, A. J. (2006). Modelling asymmetric exchange rate dependence. International Economic Review, 47(2), 527-556.