Explain the scenario, volatility and correlation z-scores along with the Mahalanobis distance, and how they feed into judging both the plausibility and the size of a scenario.

FRM® Part II · Risk Management and Investment Management · Market-Driven Scenarios: An Approach for Plausible Scenario Construction · FRM2.IM.15.b

The idea

A z-score expresses a move in standard deviations, so a monthly shock must be compared with monthly, not annual, volatility. Individual z-scores say how large each move is. For one move, z = shock ÷ volatility over the scenario horizon. The volatility z-score reflects the size of the scenario judged on volatilities alone, ignoring co-movement. These scores miss whether the combination of moves is typical. The Mahalanobis distance, written D, fixes this by using R, the correlation matrix, so it penalises combinations that break historical correlations. Here z is the vector of the individual z-scores, one for each variable. In matrix form, D = √(z′R⁻¹z). For two variables, the squared distance is written Dsq. Then Dsq = (z₁² + z₂² − 2ρz₁z₂) ÷ (1 − ρ²). Under joint normality, Dsq follows a chi-square distribution with as many degrees of freedom as variables, so its expected value equals the number of variables. Exam trap: opposite moves in highly positively correlated assets give a large D even when each z-score is modest.

Formulas

Z-score of a scenario shock

z = shock ÷ σh

shock: the move in the variable over the scenario horizon
σh: the variable's volatility over that same horizon

Match the horizon: a one-month shock is divided by monthly volatility, which is annual volatility ÷ √12 under square-root-of-time scaling.

Mahalanobis distance of a scenario

D = √(z′R⁻¹z)

z: the vector of the individual z-scores
R: the correlation matrix of the variables

Under joint normality D² is chi-square with as many degrees of freedom as variables, so its expected value equals that number. Opposite moves in highly correlated assets give a large D.

Squared Mahalanobis distance for two variables

Dsq = (z₁² + z₂² − 2 × ρ × z₁ × z₂) ÷ (1 − ρ²)

z₁: the z-score of the first variable
z₂: the z-score of the second variable
ρ: the correlation between the two variables

Take the square root for D. Moves that fit the correlation shrink the distance below the Euclidean one; moves against it enlarge it.

A practice question

From the question bank, with the reason each wrong option is wrong.

A manager stresses an equity index with a fall of 12% over one month. The index has annualised volatility of 18%. Scaling volatility with the square root of time, the z-score of the shock is closest to:

-0.67Why not: Compares the monthly move with annual volatility without scaling volatility to the one-month horizon.
-2.31 - correct
-10.58Why not: Scales annual volatility to a daily horizon (divides by the square root of 252) although the shock is monthly.
-0.19Why not: Multiplies annual volatility by the square root of 12 instead of dividing, inflating the denominator.

Monthly volatility = 18% / sqrt(12) = 5.196%. z = -12 / 5.196 = -2.31, so the shock is about 2.3 monthly standard deviations.

A harder one

Two equity indices have a correlation of 0.8. In a scenario their monthly shocks are +2 and -2 standard deviations respectively. The Mahalanobis distance of the scenario is closest to:

6.32 - correct
2.83Why not: Ignores the correlation: the Euclidean distance square root of (4 + 4).
2.11Why not: Has the sign wrong, treating the moves as if both were in the same direction (+2 and +2).
40.00Why not: Reports the squared distance, not its square root.

D² = (4 + 4 - 2 x 0.8 x (2)(-2)) / (1 - 0.64) = 14.4 / 0.36 = 40, so D = 6.32. Opposite moves in highly correlated assets are extremely unusual.

The hardest tier

A scenario has an equity shock of -15% and a credit spread shock of +60 basis points over one month. Monthly volatility is 5% for equity and 20 basis points for the spread, and their correlation is -0.6. The Mahalanobis distance is closest to:

4.24Why not: Ignores correlation: the Euclidean distance square root of (9 + 9) from the z-scores of -3 and +3.
6.71Why not: Uses a correlation of +0.6 in the formula, treating the two moves as opposite to the usual relationship.
2.68Why not: Leaves out the division by (1 - rho²): square root of (18 - 10.8) = 2.68.
3.35 - correct

z-scores are -3 and +3. D² = (9 + 9 - 2 x (-0.6) x (-3)(3)) / (1 - 0.36) = (18 - 10.8) / 0.64 = 11.25, so D = 3.35. The moves fit the negative correlation, so D is below the Euclidean 4.24.

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