4.5.1 Stress Testing from Zero

Basic Prerequisites: Probability & Measure Theory (what a probability is).

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1. Intuition & Practical Objective

This page builds the why of stress testing with no prior risk knowledge needed. The objective is one idea: a risk number computed from a probability distribution cannot tell you what happens in the tail you have not observed - so you must instead decide the tail yourself, on purpose, and measure its impact. That deliberate decision is a scenario; applying it to your positions is a stress test.

Start with the dumbest question: if VaR says we lose at most $Xwith99\%confidence,whyisthefirmstillatrisk?\ast Because"99\%confidence"isastatement\ast aboutamodelfittedtohistory\ast .Itsays:\ast withinthewindowIlookedat,lossesworsethan$X happened less than 1% of the time. It says nothing about:

1. a shock bigger than anything in the window (a 100-year flood the window never contained),
2. a shock that is ordinary in size but hits several books at once (correlations rising to 1), and
3. a shock whose joint effect the model’s independence/diversification assumptions silently forbid.

The 2008 crisis is the canonical demonstration: pre-crisis, firms ran “severe” scenarios that produced losses of no more than a quarter’s worth of earnings (BIS, Principles for Sound Stress Testing, 2009); the realized GFC loss in many books was many multiples of that. The scenarios were the wrong ones, not the VaR model. Stress testing exists to force the firm to look at the tail it did not assign a probability to.

Three steps:

1. Stress is deterministic, VaR is probabilistic. A scenario is just a vector of shocks $(\Delta F_1,\dots ,\Delta F_K)$ you choose - no likelihood, no sampling. The P&L under it is a deterministic weighted sum. This is a feature, not a flaw: it lets you study exactly the event you fear, at whatever severity you fear.
2. The point is survival, not a number. VaR/ES feed limits and capital models. A stress test answers the existential question: does this shock wipe out capital? trigger margin calls? force a fire-sale? That is why regulators (CCAR/DFAST, EBA, FRTB stressed ES) build capital around stressed losses.
3. Stress is about correlation breaking, not volatility rising. A shock that hits one asset is handled by diversification. The dangerous scenario is one where everything falls together - the diversification you priced in silently disappears.

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2. Mathematical Ground Truth & Derivations

The factor P&L model. Map the portfolio to $K$ risk factors with linear sensitivities (deltas) ${\beta }_k$ - the dollar change in portfolio value per unit move in factor $k$ (equity %, credit bps, rate bps, …). Under a shock vector $\Delta F$ the P&L is, to first order,

$$\Delta V=\sum _{k=1}^K{\beta }_k\Delta F_k.$$

This is exactly the linear/delta mapping of Hull OFOD Ch 22 (eq. 22.6, $\Delta P=\sum S_i{\delta }_i\Delta x_i$). A stress scenario is simply a chosen vector $\Delta F^{\ast }$; the stressed loss is $\Delta V^{\ast }={\beta }^T\Delta F^{\ast }$. Second-order terms (options, convexity) add $\frac{1}{2}\gamma (\Delta S{)}^2$ and cross-gammas, but the machinery is identical - stress still feeds a deterministic shock through the P&L.

Why the same shock hits different portfolios differently. The scenario’s effect is ${\beta }^T\Delta F$: a long-credit book has large negative ${\beta }_{credit}$, so the 2008 “credit +500bp” leg dominates; a long-equity book is hit by the equity leg; a matched book may be near-neutral to any single leg but exposed to the interaction. The scenario converts abstract risk into a dollar P&L per book, which is why firms run it per desk and per risk type.

Historical vs hypothetical scenarios (BIS 2009, Principles): historical scenarios replay an observed crisis (1987, LTCM, 2008, COVID) by re-applying its realized factor moves; hypothetical scenarios are constructed by the risk team to stress what history never produced (a rates spike and a credit blow-out simultaneously, a sovereign default, a liquidity freeze). Both feed the same ${\beta }^T\Delta F$ engine - they differ only in where the shock vector comes from.

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3. Computational Implementation - deterministic crisis replay

Stdlib only. Apply historical crisis shocks to a $100M long-credit, long-duration fund; the P&L is a plain weighted sum.

The 2008 replay loses $47M - 47% of NAV - on a portfolio whose VaR model, calibrated to 1995–2007, would have called this a near-impossible event. The scenario does the work; no distribution was needed.

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4. Failure Modes & First-Principles Breakdowns

1. The “probability of the scenario” trap. A beginner asks “but how likely is this scenario?” - that is the wrong question. Stress testing by design discards probability because the tail’s likelihood is unknowable (fat tails make tail probabilities poorly estimated - see Extreme Value Theory & Fat Tails). Demanding a probability forces you back into the distribution you were trying to escape.
2. Scenario is not forecast. A stress shock is a hypothesis about joint factor behavior, not a prediction of where markets go. It will almost surely not happen as specified; its value is in exposing which combination of moves threatens survival, not in predicting the move.
3. First-order linearity hides convexity. A delta-only stress misses the gamma loss: an options book loses more than $\beta \Delta F$ because the delta itself moves. Stress tests on option-heavy books must add $\frac{1}{2}\gamma (\Delta S{)}^2$ (see Black–Scholes–Merton Greeks). Neglecting this understates exactly the tail you are trying to measure.

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5. References

• BCBS: Principles for Sound Stress Testing Practices and Supervision (2009, CN14)
• Hull, Options, Futures, and Other Derivatives
• McNeil, Frey & Embrechts, Quantitative Risk Management
• Tsay, Analysis of Financial Time Series

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6. Connected Graph Bridges

• Base: Probability & Measure Theory · VaR & Expected Shortfall · Extreme Value Theory & Fat Tails
• Continue: 02 · Why Stress Testing · Index Hub