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Account for the steps in defining and building market-driven scenarios, and outline how the MDS approach is applied in the worked example.

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

The idea

Building an MDS follows a clear sequence. First define the narrative and choose the key variables to shock, the size of each shock and the horizon. Second, estimate the covariance matrix of the risk factors over a suitable window and scale it to that horizon. Third, derive the unshocked variables as conditional expectations given the shocks. For one shocked variable x, the expected move of another variable y is found with Move = ρ × (σy ÷ σx) × shock to x. With several shocks, the correlation matrix of the shocked variables must be inverted so that their mutual correlation is not double counted. Fourth, revalue the portfolio with the full set of factor moves. Finally, assess plausibility with z-scores and the Mahalanobis distance, adjusting the shocks if the scenario is too remote or too mild. In the worked example, an equity shock is propagated to rates, spreads and currencies. This is a linear, model-based approximation from the historical covariance matrix, not tail dependence, and stressed dependence is a separate choice made by shocking correlations explicitly. Exam trap: using only the correlation, without the volatility ratio.

Formulas

Conditional expected move of an unshocked variable

E(Δy) = ρ × (σy ÷ σx) × Δx

ρ
the correlation between x and y
σy
the volatility of y over the horizon
σx
the volatility of x over the horizon
Δx
the shock to x

For one shocked variable, from the historical covariance. Using the correlation without the volatility ratio is the usual trap; with several shocks their correlation matrix must be inverted.

A practice question

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

In the MDS approach, what determines the value assigned to a variable that was not directly shocked?

  1. Its conditional expectation given the shocked values, derived from the estimated covariance matrix. - correct
  2. Its unconditional historical average return, because the shocks apply only to the variables the analyst chose.Why not: Ignores the link to the shocks; the unconditional mean would also give an uncoupled scenario.
  3. Its own worst historical move over the estimation window, in whichever direction it occurred.Why not: Uses a stand-alone extreme rather than the move implied by the shocked variables.
  4. A value set by the analyst, free of any constraint from the covariance matrix.Why not: Describes an ad hoc scenario, which gives up MDS's coherence.

The unshocked variables are set to E[y | shocked variables], a regression-like result that uses correlations and volatilities. This is why the scenario stays internally consistent.

A harder one

In an MDS, equity is shocked by -25%. Equity volatility over the horizon is 20% and the volatility of credit spread changes is 100 basis points, with a correlation of -0.5. The conditional expected change in the credit spread is closest to:

  1. -62.5 basis pointsWhy not: Gets the sign wrong; a negative correlation with a fall in equity implies a spread widening.
  2. +12.5 basis pointsWhy not: Leaves out the ratio of volatilities (100 / 20) and uses rho x shock = 12.5.
  3. +125 basis pointsWhy not: Assumes a perfect correlation of -1, giving 5 x 25.
  4. +62.5 basis points - correct

Expected move = rho x (sigma_y / sigma_x) x shock = -0.5 x (100 / 20) x (-25) = +62.5 basis points.

The hardest tier

In an MDS, variables 1 and 2 are shocked by -2 standard deviations each, and their correlation is 0.5. Variable 3 has a correlation of 0.6 with variable 1 and 0.2 with variable 2, and its scenario-horizon volatility is 10%. Its conditional expected move is closest to:

  1. -16.0%Why not: Adds rho x z separately for each shocked variable, double counting because the shocked variables are themselves correlated.
  2. -12.0%Why not: Uses only variable 1's correlation (0.6 x -2 x 10%), ignoring variable 2.
  3. -10.7% - correct
  4. -8.0%Why not: Uses the average correlation of 0.4 with the shocks: 0.4 x -2 x 10%, ignoring the structure of the matrix.

z3 = rho' R^-1 z. R^-1 z = (1/0.75) x (-2 + 1, 1 - 2) = (-1.333, -1.333), so z3 = (0.6 + 0.2) x (-1.333) = -1.067 and the move is -1.067 x 10% = -10.7%.

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