4.5 Stress Testing & Scenario Analysis

Basic Prerequisites: VaR & Expected Shortfall (why risk measures work in probability) and Linear Algebra (covariance matrices, the quadratic form $z^{T} Σ^{- 1} z$ ). (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

VaR and Expected Shortfall ask “how bad is the normal tail?” - they estimate a quantile of a fitted or historical loss distribution, then treat that number as the risk. Stress testing asks a completely different question: “what happens to this firm if the unforecastable happens?” It deliberately throws away the probability distribution and instead subjects the portfolio to a deterministic shock - a replay of 2008, or a hypothetical “equities −40% and credit +500bp at the same time.”

Why bother, if we already have VaR/ES? Because probability is the wrong ruler for the rare tail. A 99% VaR computed from history is silent about the crisis day that has not happened yet: the 2008 loss in several books was many multiples of their VaR. Stress testing is the bridge from risk measurement (how much could we lose under a model) to survival analysis (can the firm keep operating when the model’s assumptions break). It is also the mechanism regulators use - CCAR/DFAST, EBA, and the FRTB’s stressed Expected Shortfall all force banks to compute capital against a stressed, not a “normal,” loss.

The one-sentence essence. “A risk measure tells you a number; a stress test tells you a path. VaR/ES ask ‘how bad is the usual tail,’ stress testing asks ‘what breaks the firm’ - and because tail probability is unknowable, the answer to the second question cannot come from a distribution, only from deliberately engineered shocks.”

2. Mathematical Ground Truth & Derivations

Quick-Reference Lookup (job #1). The formulas below are transcribed from the verified corpus - FRTB (BIS 2019, d457), Hull OFOD Ch 22, McNeil–Frey–Embrechts QRM Ch 13, Schuermann (2014) - and the numbers in the check column were re-executed and reproduced exactly (see §3).

Notation: $V$ portfolio value, $β_{k}$ sensitivity (delta) to risk factor $k$ , $Δ F_{k}$ factor shock, $Σ$ factor covariance, $z$ standardized factor shock vector, $C$ capital to breach, $ρ$ correlation.

Quantity	Formula	Verified check
Factor-model P&L	$Δ V = \sum_{k} β_{k} Δ F_{k}$	1987: $- 14.75 M$ $ on the example fund
Normal 1-day 99% VaR (single asset)	$2.326 σ$	$2.326 \times 0.015 = 3.49%$
Normal ES @ $100 (1 - α) %$	$μ + \frac{σ ϕ ( z _{α} )}{1 - α}$	99% $\Rightarrow$ ES $= 2.665 σ$ vs VaR $2.326 σ$
Reverse stress test (min shock to breach)	$min z^{T} Σ^{- 1} z$ s.t. $β^{T} z \leq - C \Rightarrow z^{*} = - \frac{C}{β ^{T} Σ β} Σ β$	normal corr: $1.866 σ$ ; stressed corr: $1.283 σ$
Min breach magnitude	$z^{* T} Σ^{- 1} z^{*} = \frac{C}{β ^{T} Σ β}$	$30 M$ $/√258.6 = 1.866 σ
FRTB stressed ES (capital measure)	$ES = ES_{R, S} \times \frac{ES _{F, C}}{ES _{R, C}} \geq ES_{R, S}$	$100 \times \frac{120}{90} = 133.3$ , ratio floored at 1
1-yr 99% VaR, 2-asset $ρ$	$2.326 w_{1}^{2} σ_{1}^{2} + w_{2}^{2} σ_{2}^{2} + 2 ρ w_{1} w_{2} σ_{1} σ_{2}$	$ρ = .3 : 37.51%$ , $ρ = .9 : 45.34%$

Scenario catalog (stylized calibration used throughout this folder; shocks are scenario inputs, not point forecasts):

Scenario	Equity	Credit spread	Rates	Typical stress lesson
1987 Black Monday	$- 20.5%$ (1 day)	+100bp	−50bp	single-day jump breaks $d t \to 0$ hedging
1998 LTCM / Russia	$- 15%$	+250bp	−150bp	flight to quality: credit blows out as rates rally
2008 GFC / Lehman	$- 40%$ (to $- 57%$ peak-to-trough)	+500bp	−100bp	simultaneous multi-asset collapse, corr $\to + 1$
2020 COVID	$- 30%$	+300bp	−50bp	liquidity vacuum: even “safe” assets sold

3. Computational Implementation - the scenario engine

This runs on the standard library only and reproduces every verified number above: deterministic factor shocks are just weighted sums, so no numpy is needed.

A $100 Ml o n g - cre d i t, l o n g - d u r a t i o n f u n d l oses$ 47M under the 2008 replay - a single deterministic calculation, no distribution required. The full machinery (sensitivity grids, scenario matrices, reverse stress, correlation breakdown, macro projection) lives in the sub-pages below.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Scenario selection bias - the result is decided by which scenarios you chose to run, not by data; two defensible scenario sets can reach opposite conclusions.
Stress has no probability - a scenario carries no likelihood, so you cannot aggregate, backtest, or rank scenarios the way you can VaR exceptions.
Correlation breakdown - the biggest tail losses come from diversification disappearing (correlations → 1), which is exactly the input most models freeze at “normal” values.

5. References

BCBS (BIS): Minimum Capital Requirements for Market Risk (FRTB, Jan 2019, d457)
BCBS (BIS): Principles for Sound Stress Testing Practices and Supervision (2009, CN14)
Federal Reserve: 2025 Stress Test Scenarios (CCAR/DFAST)
Schuermann, Til: Stress Testing Banks, IJCB 10(2):95–154 (2014)
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
McNeil, Frey & Embrechts: Quantitative Risk Management (2015)
Quagliariello (ed.) (2009), Bellini (2016)

6. Connected Graph Bridges

Foundational base: VaR & Expected Shortfall · Extreme Value Theory & Fat Tails
Sub-pages (in-folder): 01 From Zero · 02 Why Stress Testing · 03 Scenario Construction · 04 Reverse Stress Testing · 05 Failure Modes · 06 Advanced Extensions
Sibling topics: Parametric, Historical & Monte Carlo VaR · Liquidity Risk & Margin Spirals · Credit Risk & Merton

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

4.5 Stress Testing & Scenario Analysis

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the scenario engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

4.5.1 Stress Testing from Zero

4.5.2 Why Stress Testing

4.5.3 Scenario Construction

4.5.4 Reverse Stress Testing

4.5.5 Failure Modes & Real-World Practice

4.5.6 Advanced Extensions