4.5.3 Scenario Construction

Basic Prerequisites: 02 · Why Stress Testing and Linear Algebra.

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1. Intuition & Practical Objective

The objective is the craft of building scenarios. A stress test is only as good as its scenario set - the entire output inherits the judgment that went into choosing the shocks. This page lays out the four tools in order of increasing sophistication and how they combine: sensitivity analysis (one factor at a time), scenario matrices (many factors, many severities), historical scenarios (replay an observed crisis), and hypothetical scenarios (construct what history never produced).

The ladder, intuitively:

1. Sensitivity (single-factor) analysis - the most basic tool: shock one input (equity −10/20/30%) and read the P&L. Cheap, fast, reveals per-factor exposure and concentration. BIS 2009 calls these “the most basic level… [their] main benefit is that they can provide a fast initial assessment of portfolio sensitivity to a given risk factor and identify certain risk concentrations.”
2. Scenario matrix - shock several factors simultaneously across a grid. Because factors interact, testing them one at a time “may not reveal their potential interaction (particularly if that interaction is complex and not intuitively clear)” (BIS 2009). The matrix is the workhorse: rows = severities, columns = factor combos.
3. Historical scenarios - replay a real crisis (1987, LTCM, 2008, COVID) using its realized joint moves. Their virtue: realistic joint dynamics. Their weakness (BIS 2009): they cannot cover new products or severities beyond the observed episode - 2008’s loss exceeded any previous historical replay.
4. Hypothetical scenarios - designed by the risk team for events history has not produced: a sovereign default, a simultaneous rates-spike + credit-blow-out, a liquidity freeze, a correlation jump. This is where judgment (and, post-2008, supervisory guidance on “severely adverse but plausible”) enters.

The one-sentence essence. “Build scenarios from bottom-up sensitivities to find where the risk is, then top-down hypotheticals to decide what you are willing to lose on; historical replays give realistic joints, hypotheticals give imagination - and a scenario that cannot lose you money is a scenario not worth running.”

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2. Mathematical Ground Truth & Derivations

Sensitivity = first derivative. The sensitivity of portfolio value to factor $k$ is the delta ${\beta }_k=\partial V/\partial F_k$. A one-factor sensitivity test applies $\Delta V\approx {\beta }_k\Delta F_k$ for a ladder of shocks $\Delta F_k\in \{-10\%,-20\%,-30\%,-40\%\}$ (equity) or $\{+100,+300,+500\text{bp}\}$ (credit). It is exact only if the P&L is linear in that factor; for convex books the second-order term $\frac{1}{2}\gamma (\Delta S{)}^2$ must be added.

Scenario matrix = tensor of factor states. A two-factor matrix is the outer product of shock sets $\{\Delta F_1^{(i)}\}\times \{\Delta F_2^{(j)}\}$, evaluated at each cell:

$$\Delta V_{ij}={\beta }_1\Delta F_1^{(i)}+{\beta }_2\Delta F_2^{(j)}+{\beta }_{12}\Delta F_1^{(i)}\Delta F_2^{(j)}+\dots$$

The ${\beta }_{12}$ cross term captures statistical interaction; the worked example below uses ${\beta }_{12}=0$, so its dominating cell is the joint-occurrence corner where both factors move badly together - exactly the correlation-breakdown the matrix exists to expose (the example demonstrates simultaneity, not curvature).

Historical scenario = empirical joint draw. A historical scenario replays the realized vector $\Delta F^{(\text{crisis})}$ from a specific date range. It is not a random draw from the unconditional return distribution; it is the joint realization of that period, so it automatically carries the realized correlations (which during a crisis → 1). Formally it is a single point in factor space, picked from history rather than constructed.

Hypothetical scenario = constructed point. The team specifies each $\Delta F_k$ by judgment (informed by historical extremes + economic logic). The craft constraint is plausibility: the shocks should be mutually consistent with an underlying story (a “2008-like” recession implies equity, credit, and rates moving in the coherent directions, not arbitrary opposites). BIS 2009 emphasizes scenarios must be “severe but plausible.”

Choosing severity. A common discipline is to calibrate the hypothetical to the historical worst plus a margin: take each factor’s realized worst stress (e.g. equity $-57\%$ peak-to-trough 2008), then construct a “worse than 2008” version to test capital adequacy - this is the logic behind CCAR’s severely-adverse scenario (see 06).

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3. Computational Implementation - sensitivity grid + scenario matrix

Stdlib only. First a one-factor sensitivity grid on the example fund, then a two-factor scenario matrix (equity × credit) that exposes the interaction corner.

Reading the matrix: equity-only sensitivity tops out at $-$20M, but the corner cell (equity $-30\%$ and credit +500bp) is $-$45M - more than twice the single-factor worst. A one-at-a-time sensitivity study would have called the fund’s risk “$-$20M”; the matrix reveals the real tail is the joint shock. This is why regulators require multi-factor scenarios, not just ladders.

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4. Failure Modes & First-Principles Breakdowns

1. Interaction blindness (single-factor-only). Testing factors one at a time misses cross terms - as the matrix above shows, the joint corner dominates. BIS 2009 flags this explicitly: scenarios must be run together because the interaction is “complex and not intuitively clear.”
2. Historical-scenario stagnation. Replaying only observed crises guarantees you are always preparing for the last war. BIS 2009: historical scenarios “were not able to capture risks in new products,” and their “severity levels and duration… proved to be inadequate” in 2008. Historical scenarios must be augmented with hypothetical severity margins.
3. Plausibility blindness (hypothetical-scenario side). The opposite failure: inventing arbitrary shocks that violate economic logic (e.g. equities −80% with credit tightening). An implausible scenario wastes effort and is dismissed by management. The discipline is internal consistency: every scenario should have a coherent story tying its shocks together.
4. Severity ratchet-up failure. Even when teams built hypotheticals pre-2008, they were “generally only moderate,” and “it was difficult for risk managers to obtain senior management buy-in for more severe scenarios” (BIS 2009). The governance failure - not the math - was the binding constraint. Regulators answered with supervisory scenarios (CCAR/EBA) precisely because firms would not self-inflict severity.

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5. References

• BCBS, Principles for Sound Stress Testing Practices and Supervision (2009, CN14)
• Hull, OFOD
• McNeil, Frey & Embrechts, QRM
• Bellini, Stress Testing and Risk Integration in Banks (2016)

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6. Connected Graph Bridges

• Back: 02 · Why Stress Testing · Index Hub
• Continue: 04 · Reverse Stress Testing
• Sibling: Parametric, Historical & Monte Carlo VaR (the historical-simulation side of “scenarios from data”)