FRM Exam Part II · Market-Driven Scenarios: An Approach for Plausible Scenario Construction

Conditional Expected Shocks and Mahalanobis Distance Explained

Updated 11 October 2026 · Fact-checked

Conditional expected shocks tell you how non-stressed risk factors should move when one factor is stressed. Under joint normality, the expected shock to factor j equals the correlation times the ratio of volatilities times the stress: ρ × (σj ÷ σi) × s. The Mahalanobis distance measures how unlikely the whole scenario is.

Understand Conditional Expected Shocks and Factor Correlations

A stress test often shocks one factor, such as equities down 15%. But other factors do not sit still. If equities fall, credit spreads usually widen and some currencies weaken. If you leave those factors at zero, the scenario is unrealistic. If you pick their moves by judgement, the scenario is hard to defend.

The market-driven approach lets the data decide. You take the covariance matrix of the risk factors and ask: given the shock to the stressed factor, what is the expected move in each other factor? For two factors with zero mean, this is a regression. The expected move in factor j is the beta of j on i, times the shock. Beta equals covariance ÷ variance of the stressed factor, which is the same as ρ × σj ÷ σi.

This result is exact when the factors are jointly normal. With several stressed factors you use the matrix form: the expected shocks to the free factors depend on the covariance between free and stressed factors, and on the inverse of the covariance of the stressed factors. The expected shock is the centre of the conditional distribution, so the other factors can still move around it. That spread is the conditional variance, and it is smaller than the unconditional variance whenever correlation is not zero.

The Mahalanobis distance answers a different question: how plausible is the whole scenario? It measures the distance of the scenario from the mean, scaled by the covariance matrix, so it accounts for both volatility and correlation. A larger distance means a less likely scenario. Under normality, the squared distance follows a chi-square distribution with degrees of freedom equal to the number of factors.

The link between the two ideas is useful. Of all scenarios that contain the given stress, the conditional expected scenario has the smallest Mahalanobis distance. So it is the most likely scenario consistent with the stress. Any other choice for the free factors is less plausible.

Key formulas to remember

Beta of factor j on stressed factor i: βji = σij ÷ σi² = ρij × σj ÷ σi; σij is the covariance. This is the slope of j on i.
Conditional expected shock (one stressed factor, zero means): E[Rj | Ri = s] = ρij × (σj ÷ σi) × s; Exact under joint normality. Use shocks and volatilities over the same horizon.
Conditional expected shock (general means): E[Rj | Ri = s] = μj + ρij × (σj ÷ σi) × (s − μi); With zero or negligible means, this reduces to the previous formula.
Conditional variance: Var(Rj | Ri) = σj² × (1 − ρij²); It does not depend on the size of s under normality.
Multi-factor conditional mean: E[x2 | x1 = s] = μ2 + Σ21 × Σ11⁻¹ × (s − μ1); x1 is the stressed set, x2 is the free set. Σ11 is the covariance of stressed factors. Σ21 is the covariance between free and stressed factors.
Multi-factor conditional covariance: Σ22|1 = Σ22 − Σ21 × Σ11⁻¹ × Σ12; Uncertainty left in the free factors after the stress is set.
Mahalanobis distance: M² = (x − μ)ᵀ Σ⁻¹ (x − μ); x is the scenario vector. Under normality, M² follows a chi-square with n degrees of freedom.
Two-factor Mahalanobis distance: M² = (z1² − 2ρ z1 z2 + z2²) ÷ (1 − ρ²), where zk = (xk − μk) ÷ σk; For a single factor, M = |z|.

How to solve Conditional Expected Shocks and Factor Correlations questions

Use this method for any question that asks for implied shocks, conditional moves or the plausibility of a scenario.

1Identify the stressed factor or factors and the shock size. Check the horizon of the shock matches the horizon of the volatilities.
2Collect the inputs: volatilities, correlations or covariances, and means if given. If only correlations and volatilities are given, covariance = ρ × σi × σj.
3For one stressed factor, compute the beta: ρ × σj ÷ σi. Put the volatility of the factor being predicted on top.
4Multiply beta by the shock (minus the mean if the mean is not zero) to get the expected shock for each free factor.
5If asked for uncertainty, compute the conditional standard deviation: σj × √(1 − ρ²).
6If asked for plausibility, standardise each move into z-scores and compute M² with the covariance inverse. For one factor, M = |s| ÷ σ.
7Interpret: the expected shock is the most likely value given the stress, not a worst case. A large M means a rare scenario. Compare M² with chi-square values when asked about probability.

Quickest way: Beta times shock, then z-scores

When to use it: Use for two-factor questions with volatilities and a correlation, which is the usual MCQ format.

Write the volatility ratio first: σ of the free factor ÷ σ of the stressed factor.
Multiply by ρ and then by the shock. Check the sign matches the sign of ρ.
Sanity check: with ρ = 1 the free factor moves by the volatility ratio times the shock. With ρ = 0 it does not move at all. Your answer must lie between these.
For Mahalanobis in two factors, convert both moves to z-scores and use (z1² − 2ρ z1 z2 + z2²) ÷ (1 − ρ²). If the second move is already the conditional expectation, M² is just z1².

Common mistakes in Conditional Expected Shocks and Factor Correlations

Inverting the volatility ratio and using σi ÷ σj
Students remember ρ and the ratio but not which factor sits on top.
Fix: The factor you are predicting goes on top. Think of beta as covariance ÷ variance of the factor that is shocked.
Using covariance directly as the shock multiplier
Covariance looks like the link between the two factors.
Fix: Divide covariance by the variance of the stressed factor. Covariance ÷ σi² is the beta.
Treating the conditional expected shock as the worst case for the free factor
The word 'shock' suggests an extreme outcome.
Fix: It is the conditional mean. The free factor still has a spread of σj × √(1 − ρ²) around it.
Computing Mahalanobis distance without the inverse covariance, or without correlation
Students add squared z-scores and treat the factors as independent.
Fix: Use Σ⁻¹. In two factors, include the −2ρ z1 z2 term and divide by (1 − ρ²). Adding z² values is correct only when ρ = 0.
Reading M² as a probability
A larger number feels like a percentage.
Fix: M² is a distance. Under normality, compare it with a chi-square distribution with n degrees of freedom to get a probability. Larger means less likely.
Assuming historical correlations hold in the stress
The method uses one estimated matrix, so the output looks precise.
Fix: State the limit. The approach assumes normality and a stable covariance matrix. Correlations often rise in crises, so the implied shocks may be understated.

Worked examples

Example 1

Monthly volatilities are 5% for an equity index and 2% for EUR/USD. The correlation is 0.4. Means are taken as zero. If the equity index falls 15% in a month, what is the expected EUR/USD move, using a multivariate normal model? Options: (A) −2.4% (B) −6.0% (C) −15.0% (D) −1.0%

Show the solution

Stressed factor is equity, with s = −15%. Free factor is EUR/USD.
Beta = ρ × σEURUSD ÷ σequity = 0.4 × 2% ÷ 5% = 0.4 × 0.4 = 0.16.
Expected shock = 0.16 × (−15%) = −2.4%.
Check: option B (−6.0%) is 0.4 × 15% and skips the volatility ratio. Option C assumes the factors move one for one.

Answer: (A) −2.4%

Example 2

Two factors have monthly volatilities of 4% and 3% and a correlation of 0.5. Means are zero. Factor 1 is stressed to −8%. (a) Find the conditional expected move in factor 2 and its conditional standard deviation. (b) Compute the Mahalanobis distance M of the scenario (−8%, −3%). (c) Compare with the scenario (−8%, 0%).

Show the solution

(a) Beta = 0.5 × 3% ÷ 4% = 0.375. Expected move = 0.375 × (−8%) = −3%.
Conditional standard deviation = 3% × √(1 − 0.25) = 3% × 0.866 = about 2.60%.
(b) z1 = −8 ÷ 4 = −2. z2 = −3 ÷ 3 = −1.
M² = (z1² − 2ρ z1 z2 + z2²) ÷ (1 − ρ²) = (4 − 2 × 0.5 × 2 + 1) ÷ 0.75 = 3 ÷ 0.75 = 4. So M = 2.
This equals |z1| = 2, as expected, because factor 2 is already at its conditional mean.
(c) For (−8%, 0%): z2 = 0, so M² = 4 ÷ 0.75 = 5.33 and M = about 2.31.
(−8%, 0%) has a larger distance, so it is less plausible than (−8%, −3%).

Answer: (a) Expected move −3%, conditional standard deviation about 2.60%. (b) M = 2 (M² = 4). (c) M = about 2.31, so leaving factor 2 unchanged is less plausible.

Exam tips

Write the volatility ratio with the predicted factor on top before touching the numbers. Wrong-ratio options are almost always among the choices.
Check the sign and size. The shock to the free factor has the sign of ρ times the stress, and its magnitude is less than the stress times the volatility ratio unless ρ = 1.
If the question gives a scenario in which the free factors are already at their conditional means, the Mahalanobis distance equals the standardised stress alone. Do not invert a matrix.
For probability questions, remember M² is chi-square with n degrees of freedom under normality, and larger M means less likely.
Expect interpretation questions on limits: normality, stable correlation and the fact that the conditional mean is not a worst case.

Practice questions from Market-Driven Scenarios: An Approach for Plausible Scenario Construction

Conditional Expected Shocks and Factor Correlations: frequently asked questions

How do I calculate implied shocks from a correlation matrix?

Take the stressed factor's shock and multiply it by ρ × σj ÷ σi for each other factor j. This is the conditional expectation under joint normality. You need volatilities as well as correlations.

What does the Mahalanobis distance show in stress testing?

It measures how far a scenario is from the mean after adjusting for volatilities and correlations. A larger distance means a less likely scenario. Under normality, its square follows a chi-square distribution with one degree of freedom per factor.

Why is the conditional expected scenario called the most likely scenario?

Among all scenarios that contain the given shock, it has the smallest Mahalanobis distance. Any other values for the free factors are further from the centre of the distribution, so they are less probable.

Does the conditional shock depend on the size of the stress?

The expected shock scales linearly with the stress. The conditional variance of the free factors does not change with it under normality. So a larger stress moves the centre but not the spread.