4.5.2 Why Stress Testing

Basic Prerequisites: 01 · VaR & ES From Zero and Parametric, Historical & Monte Carlo VaR.

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1. Intuition & Practical Objective

The task is to state precisely why a good risk measure is still not enough - and where the boundary between risk measurement and stress testing lies. The objective is a discipline, not an argument: you must know which question each tool answers, because using VaR to answer the survival question is a category error. Risk measures estimate a distributional quantile from observed behavior; stress testing studies the behavior of a specific portfolio under a specified shock without claiming the shock’s probability.

Three concrete ways VaR/ES fail to be the whole story:

1. The tail in the window ≠ the tail in the world. VaR is a statement about the sample (or fitted model) you fed it. A 99% daily VaR from 1995–2007 did not contain 2008’s moves. This is not a bug in VaR - it is a fundamental limit: you cannot estimate the probability of an event you have never observed and whose generating process is not stationary. Stress testing sidesteps the estimation problem entirely by not requiring a probability.
2. ES averages a tail the model thinks is normal. Expected Shortfall fixes VaR’s subadditivity flaw (see Coherent Risk Measures) and is more sensitive to tail shape - but it still averages losses within the model’s tail, whose shape is set by a normal/student/GPD fit. If the true tail is fatter or the joint tail different (correlations rise), ES is systematically too small. Stress testing applies shocks the fitted tail never contemplated.
3. Risk measures are about normal behavior by construction. VaR/ES calibrate to the current/typical regime. The FRTB’s answer - stressed ES - is itself a concession that a plain ES calibrated to recent data is not a capital-adequate tail measure: regulators force the ES to be re-calibrated to a 12-month stress period (see 06 · Advanced Extensions).

The one-line boundary. “VaR/ES answer ‘how bad is the tail under my model?’ - stress testing answers ‘what happens to my book under the shock I fear?’ - the former needs data and an assumed distribution, the latter needs neither, only a scenario.”

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2. Mathematical Ground Truth & Derivations

Why the quantile cannot see the unseen tail. Let losses be $L$, with 99% daily VaR $q={\text{VaR}}_{0.99}(L)$. For any distribution, ES is the average beyond $q$:

$${\mathrm{ES}}_{0.99}(L)=\mathbb{E}[L\mid L>q].$$

For a normal $L\sim \mathcal{N}(\mu ,{\sigma }^2)$ the ratio is constant: $\mathrm{ES}/\mathrm{VaR}=\frac{\varphi (z_{0.99})}{(1-0.99)z_{0.99}}$ - with $z_{0.99}=2.326$, $\varphi (2.326)=0.02665$, this ratio is $\approx 2.665/2.326=1.146$. So for a normal world, ES is VaR plus ~15%. But the ratio is a property of the assumed distribution. For a fat-tailed (e.g. Student-$t$ or GPD) tail, ES can be a multiple of VaR - the whole point of EVT (EVT & Fat Tails). Neither VaR nor ES tells you about a joint shock whose correlation structure is wrong.

Where stress enters. A stress scenario fixes a shock vector $\Delta F^{\ast }$ and evaluates

$$\Delta V^{\ast }={\beta }^T\Delta F^{\ast }+\frac{1}{2}\Delta F^{\ast T}\Gamma \Delta F^{\ast }+\dots$$

This is not a quantile of anything - it is a point evaluation. Its information is orthogonal to VaR’s: it tells you the portfolio’s behavior in one specific state, which VaR (an average over many states) necessarily obscures. The most important such state is the correlation-breakdown state, where the covariance that VaR uses is itself wrong - treated in 05 · Failure Modes.

A sharp numerical demonstration (reproduced exactly in §3): on a normal sample, the 1-day 99% VaR and ES of the equity book are small - but a single crisis day produces a loss ~5× ES, because the crisis day sits outside the normal tail the measure summarized. That ratio (VaR/ES vs realized stress loss) is the empirical signature of “risk measure calibrated to the wrong regime.”

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3. Computational Implementation - normal VaR/ES vs a crisis day

Stdlib only. Draw a normal sample, compute historical 99% VaR and ES, then drop a single Black-Monday-style day on top - the stress loss is a multiple of the measure.

The 99% VaR ($1.86M)andES($2.02M) - both perfectly valid quantile estimates of the normal sample - are crushed by a single realized crisis day ($10.25M). The measure was right about its own distribution and useless about the regime that actually produced the loss. This is the case for stress testing and, at the regulatory level, for FRTB’s stressed ES.

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4. Failure Modes & First-Principles Breakdowns

1. Confusing “model tail” with “real tail.” The most common error: “our 99.9% VaR covers everything.” It covers everything the fitted distribution contains. A fat tail makes the 99.9% quantile extremely hard to estimate (huge variance) - so the claim is strongest exactly where the evidence is weakest. Stress testing does not fix estimation; it bypasses it.
2. ES-only complacency. Because ES is coherent and “better than VaR,” teams sometimes stop there. But ES is still a single scalar from one distribution; it cannot represent a scenario-specific joint stress, and it still depends on the correlation matrix that breaks under stress. Stressed ES in FRTB exists precisely because plain ES was judged insufficient for capital.
3. VaR/ES do not model the path. Both are one-horizon numbers. Stress matters over a sequence: a $47M stress loss on day 1 triggers margin calls and forced selling that become a funding crisis (see Liquidity Risk & Margin Spirals). The scenario’s path - not just its one-period P&L - is what threatens survival.

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5. References

• BCBS (BIS), Minimum Capital Requirements for Market Risk (FRTB, 2019, d457)
• Artzner et al. (1999)
• Hull, OFOD
• McNeil, Frey & Embrechts, QRM

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6. Connected Graph Bridges

• Back: 01 · From Zero · VaR & Expected Shortfall · Extreme Value Theory & Fat Tails
• Continue: 03 · Scenario Construction · Index Hub
• Forward: 06 · Advanced Extensions (FRTB stressed ES, CCAR)