4.5.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice and Credit Risk & Merton (default/loss modelling).

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1. Intuition & Practical Objective

The firm-level stress tests of 01–05 treat the portfolio as a black box and shock market factors. Macro (system-wide) stress testing takes the view up a level: it shocks macroeconomic variables (GDP, unemployment, house prices, rates) and projects their effect through to credit losses and capital ratios - because the catastrophic scenarios are the ones that are macro-driven, not just market-driven. Supervisory stress tests (CCAR/DFAST in the US, EBA in Europe) are the regulators running exactly this on banks, in public, and setting capital from the results.

The objective: understand how a single macro shock - say, the 2025 CCAR severely-adverse scenario, unemployment +5.9pp to 10% and real GDP −7.8% - flows through a loan book into PDs, expected losses, and finally into whether a bank’s post-stress capital ratio stays above the regulatory minimum. And, on the market-risk side, how the FRTB’s stressed Expected Shortfall operationalizes “capital against the stress period, not the current period.”

Three ideas, in one line each:

1. Macro-to-loss linkage. You need a model that maps $\Delta \text{GDP},\Delta u$ to probabilities of default and loss given default; the stress shock enters at the top (macro) and exits at the bottom (capital ratio).
2. Capital adequacy is the object. A stress test is not “how much do we lose” but “does post-stress capital breach the minimum?” - the binding supervisory question since 2008.
3. Stressed ES is the market-risk analogue. FRTB requires the ES to be re-calibrated to a 12-month stress period (back to 2007), not the current window - regulatory codification of everything 02 argued about “calibrated to the wrong regime.”

The one-sentence essence. “Macro stress testing turns a top-down economic scenario into a bottom-up capital ratio - and because firms would not stress themselves hard enough, supervisors now pick the scenario and publish the result.”

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2. Mathematical Ground Truth & Derivations

Macro→credit-loss linkage. Let a loan book have exposure-at-default $\text{EAD}$, loss-given-default $\text{LGD}$, and a base probability of default ${\text{PD}}_0$. Map the macro scenario to a stressed PD via an exponential/exponential-logit factor model:

$${\text{PD}}_{stress}={\text{PD}}_0\exp (a\Delta u+b|\Delta g|),$$

where $\Delta u$ is the unemployment-rate increase (percentage points) and $\Delta g$ the GDP decline (%). Expected credit loss is then

$$\text{EL}={\text{PD}}_{stress}\times \text{LGD}\times \text{EAD}.$$

Adding any market/trading loss ${\text{L}}_{mkt}$ gives total loss ${\text{L}}_{tot}=\text{EL}+{\text{L}}_{mkt}$, and post-stress capital and capital ratio follow:

$${\text{CET1}}_{post}={\text{CET1}}_0-{\text{L}}_{tot},{\text{CET1 ratio}}_{post}=\frac{{\text{CET1}}_{post}}{\text{RWA}}.$$

The bank “fails” if the post-stress ratio drops below the regulatory minimum (e.g. 4.5% CET1) - the entire point of the supervisory exercise. (This is the structure behind CCAR/DFAST: the Fed publishes the macro scenario, banks project the losses, and the Fed compares projected capital ratios against thresholds. Schuermann 2014.)

FRTB stressed Expected Shortfall. Under the Internal Models Approach, the market-risk capital measure is a stressed ES, computed from a reduced set of risk factors $R$ over the most severe 12-month period of stress (back to 2007), then scaled by the ratio of full-factor to reduced-factor current ES (floored at 1) (FRTB §33.6):

$$\mathrm{ES}={\mathrm{ES}}_{R,S}\times \frac{{\mathrm{ES}}_{F,C}}{{\mathrm{ES}}_{R,C}},\frac{{\mathrm{ES}}_{F,C}}{{\mathrm{ES}}_{R,C}}\ge 1,$$

with ES at a 97.5% one-tailed confidence, daily, and liquidity-horizon-scaled (10–120 days, §33.4). The reduced factor set must explain ≥75% of the full-model ES variation (§33.5). The math is the same stressed-ES machinery as VaR & ES - the difference is which window the ES is calibrated to: the stress period, not the current one.

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3. Computational Implementation - macro stress: does capital breach?

Stdlib only. A $10Bloanbook(RWA$8B, CET1 $1.0B,basePD1.2\%,LGD45\%)plusa$200M trading equity book, run under the baseline and the CCAR-style severely-adverse scenario (unemployment +5.9pp, real GDP −7.8%, equity −50%).

The severely-adverse scenario pushes the macro-driven PD from 1.4% to 13.4%, producing $0.70Boftotallossthatdrivespost-stressCET1to$0.30B - a 3.73% ratio, below the 4.5% minimum. The bank fails the stress test. Note the market leg (equity −50%) is the small part; the macro-to-credit linkage is what breaks capital. This is why supervisors insist on macro scenarios: the killer is credit losses, invisible to a market-only factor stress.

FRTB stressed ES, worked numerically. If the stress-period reduced-factor ES is ${\mathrm{ES}}_{R,S}=100$, the current full-factor ES ${\mathrm{ES}}_{F,C}=120$, and the current reduced-factor ES ${\mathrm{ES}}_{R,C}=90$, then $\mathrm{ES}=100\times \frac{120}{90}=133.3$. The ratio $120/90=1.33>1$ exceeds the floor of 1, so the full-factor ES’s richer risk coverage raises the capital charge above the raw stressed ES - the scaling ensures the reduced-factor stress period is not artificially cheapened.

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4. Failure Modes & First-Principles Breakdowns

1. Model risk in the macro→PD link. The exponential map is a model with its own error; a wrong slope $a,b$ (calibrated on a benign sample) understates exactly the deep-recession PDs the stress is meant to capture. This is model risk (the risk that a model is wrong in an unmeasurable way) compounded by stress: the map is extrapolated into a regime it never observed.
2. Reverse-causality / feedback loops. A macro stress that lowers capital can itself trigger deleveraging, fire-sales, and further macro decline (the Brunnermeier–Pedersen loss/funding spiral; Liquidity Risk). Most bank-level stress tests are static (no feedback), so they understate the second round. Supervisors partially compensate with system-wide exercises (e.g. the 2009 SCAP, EBA EU-wide tests).
3. Stressed-ES window selection. FRTB lets the bank pick the 12-month stress period, subject to the reduced-set ≥75% variation rule; different windows give different ES. Window- and reduced-set choice is itself a scenario-selection-bias form of model risk - the regulator constrains it but cannot eliminate it.
4. Severity ratchet and gaming. Because post-stress capital determines payouts, banks have incentive to build models that make stress losses look small (a governance failure, not a math one). This is precisely why CCAR/DFAST run the models in parallel and impose capital floors - the structure exists to counteract the bias that BIS 2009 documented.

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5. References

• Federal Reserve: 2025 Stress Test Scenarios (CCAR/DFAST)
• BCBS (BIS), Minimum Capital Requirements for Market Risk (FRTB, 2019, d457)
• Schuermann, Stress Testing Banks, IJCB 10(2) (2014)
• Quagliariello (ed.), Stress-testing the Banking System (2009); Bellini, Stress Testing and Risk Integration in Banks (2016)
• Brunnermeier & Pedersen (2009), Brunnermeier (2009)

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · Index Hub
• Sibling: Credit Risk & Merton (the default/loss modelling under the macro PD) · VaR & Expected Shortfall (ES, ESF/ESR scaling) · Liquidity Risk & Margin Spirals (the feedback loop)
• Forward: Pillar 4 · Quantitative Risk (capital & model-risk layers that stress feeds into) · Liquidity Risk (the funding feedback a macro loss triggers)