Account for the difficulties of building stress test scenarios, notably keeping risk factors coherent when they are modeled.
The idea
A stress scenario must be coherent. The risk factors in it have to move together in a way the economy and markets could actually produce. Shocking each factor to its own worst value and adding the losses ignores co-movement, and the worst moves never happened together. Historical scenarios are coherent by construction but may not be relevant to today's portfolio. Hypothetical ones are relevant but coherence is harder. A common fix is to shock a few key factors and set the others at their conditional expectation. Let X be a shocked factor and Y another factor, with correlation ρ, means μX and μY and standard deviations σX and σY. For jointly normal factors, the conditional expectation of Y given a shock x to X is Target = μY + ρ × (σY ÷ σX) × (x − μX). Plausibility can be measured by the Mahalanobis distance, which accounts for correlation. Models fitted in calm periods may also break down when extrapolated to extreme macro paths.
Formulas
Conditional expectation of an unshocked factor
E[Y | X = x] = μY + ρ × (σY ÷ σX) × (x − μX)
- μY
- the mean of the unshocked factor Y
- ρ
- the correlation between X and Y
- σY
- the standard deviation of Y
- σX
- the standard deviation of the shocked factor X
- x
- the shocked value of X
- μX
- the mean of X
Holds for jointly normal factors. Watch the sign of ρ: a negative correlation with a fall in X gives a rise in Y.
Mahalanobis distance of a two-factor scenario
d² = (x1² − 2 × ρ × x1 × x2 + x2²) ÷ (1 − ρ²)
- x1
- the first factor's move in standard deviations
- x2
- the second factor's move in standard deviations
- ρ
- the correlation between the two factors
Moves that agree with the correlation are closer to the mean than their size suggests. Take the square root for d.
A practice question
Why must the risk factors in a stress scenario be coherent?
- Coherence guarantees that the scenario has a stated probability, such as one in 100.Why not: Coherence concerns internal consistency, not probability; stress scenarios generally carry no stated probability.
- Only factors that are uncorrelated in normal times can be combined into one scenario.Why not: Reverses the point: correlated factors are the ones that must be shocked consistently.
- Factors co-move through economic and market links, so independent shocks can give an impossible scenario. - correct
- Regulators require every scenario to be based on a past historical episode, because only history is acceptable evidence.Why not: Hypothetical scenarios are allowed; history is only one source and does not by itself ensure relevance.
A scenario is a joint outcome. If equity falls sharply but credit spreads stay flat, the combination contradicts how markets behave, so losses computed from it are not meaningful.
A harder one
A scenario shocks equity returns to minus 50 percent, which is 2 standard deviations (sigma = 25 percent, mean zero). Credit spread changes have sigma = 0.60 percentage points and mean zero, with correlation minus 0.5 to equity returns, and the factors are jointly normal. What spread change should be assigned to be coherent with the equity shock (closest to)?
- Minus 0.60 percentage pointsWhy not: Sign error: ignores that spreads widen when equities fall (negative correlation).
- Plus 1.20 percentage pointsWhy not: Ignores the correlation of 0.5 and moves spreads by the full 2 sigma, as if correlation were perfect.
- Plus 0.30 percentage pointsWhy not: Multiplies by rho² (0.25) instead of rho, understating the conditional move.
- Plus 0.60 percentage points - correct
E[spread change | equity = -50%] = rho x (sigma spread / sigma equity) x shock = -0.5 x (0.60 / 25) x (-50) = +0.60 percentage points.
The hardest tier
A bank links loan loss rates linearly to unemployment and house prices using 2003-2007 data. A severe scenario, whose macro paths are coherent, puts unemployment far above any value in the sample. What is the main remaining difficulty?
- Coherence cannot be assessed whenever more than one macro variable is used.Why not: The macro paths are stated to be coherent; the issue is the loss model, not the scenario's consistency.
- Loss rates must be independent of unemployment for the model to be valid.Why not: The model relies on dependence between losses and macro drivers; independence would make the scenario irrelevant.
- Fitted relationships must be extrapolated beyond the sample range, where they may break. - correct
- The scenario is inadmissible because stress scenarios must stay inside the range of the historical sample data.Why not: Scenarios may exceed history; the real concern is that the estimated model may not hold outside the data.
Calibrating on a calm sample and applying severe, out-of-range inputs relies on linear, stable relationships that may fail in the tail. Coherent macro paths do not fix this model risk.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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