Market-Driven Scenarios: An Approach for Plausible Scenario Construction: formula sheet

Key formulas

Scenario P&L (linear approximation)

ΔP ≈ Σ (exposure_i × shock_i)

Add exposure times shock across risk factors. For options and other nonlinear positions, use full revaluation, not this approximation.

VaR definition

P(Loss > VaR) = 1 − confidence level

VaR is a threshold. It gives no information on the size of losses beyond it.

Scenario versus VaR

VaR = statistical, probability-based; scenario = event-based, usually no probability

Use this contrast to pick the right tool in a question.

Implied move for one peripheral factor (single core factor, joint normal)

E[Y | X = x] = μY + ρ × (σY ÷ σX) × (x − μX)

With zero means, implied move = ρ × (σY ÷ σX) × shock to X. Use it to infer a peripheral factor Y from a core shock X.

Regression beta form

β = ρ × σY ÷ σX = Cov(X, Y) ÷ Var(X)

Implied move in Y equals β times the shock in X (zero-mean case).

Shock size in standard deviations

z = shock ÷ σX

Measures how extreme the core shock is. Larger z means a less likely scenario.

Multi-factor conditional expectation

E[Y | X = x] = μY + Σ_YX × Σ_XX⁻¹ × (x − μX)

Σ_YX is the covariance between peripheral and core factors; Σ_XX is the core factors' covariance matrix. Applies under joint normality.

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|.

Linear P&L for one position

P&L ≈ Sensitivity × Factor shock

For a bond, P&L ≈ −DV01 × Δy in basis points. For equities, P&L ≈ Delta × market value × % shock.

Portfolio scenario P&L

P&L(scenario) = Σ over positions and factors of (sensitivity × shock)

Add across all factors hit in the scenario. Gains in hedges offset losses.

Second-order adjustment

P&L ≈ Δ × ΔS + ½ × Γ × (ΔS)²

Use for options. Convexity plays the same role for bonds: ½ × Convexity × (Δy)² × value.

Conditional expected shock

E[shock to Y | shock to X] = β × shock to X, where β = ρ × σY ÷ σX

This is the linear (normal-distribution) way to infer other factor moves in a market-driven scenario.

Stress loss versus limit

Utilisation = Scenario loss ÷ Stress limit

A utilisation above 100% means a breach and calls for action.

Quick revision

• A stress test examines losses under severe but plausible conditions that VaR may not capture.
• A market-driven scenario shocks chosen factors and infers the rest from market relationships.
• Conditional expected shock of j given a shock to i = shock × ρij × (σj ÷ σi).
• Zero correlation implies an expected move of zero in the other factor, under this approach.
• A negative correlation gives an inferred move in the opposite direction to the shock.
• Plausibility is about how likely the joint move is, not just the size of one shock.
• Larger shocks in more factors need not be more plausible; consistency with correlations matters.
• Scenario loss = sum of exposure × factor move across all factors.
• The method relies on historical volatilities and correlations, which can break down in a crisis.
• Scenarios are a complement to VaR and expected shortfall, not a replacement.
• Check units: percent moves, basis points and currency exposures must be consistent.

Common mistakes

• Treating a scenario loss as a worst-case loss. Fix: Remember that a scenario is one chosen event. A worse event can exist.
• Saying VaR tells you the average loss when VaR is exceeded. Fix: VaR is a threshold. Expected shortfall is the average loss beyond it.
• Using ρ × shock without the volatility ratio Fix: Always multiply by σY ÷ σX. Correlation only sets the sign and strength.
• Inverting the volatility ratio Fix: Y is the factor being inferred, so σY goes on top and σX, the shocked factor, goes below.
• Inverting the volatility ratio and using σi ÷ σj 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 Fix: Divide covariance by the variance of the stressed factor. Covariance ÷ σi² is the beta.
• Getting the sign wrong for bonds and short positions Fix: Write the exposure sign first. A yield rise is a loss for a long bond holder and a gain for a short.
• Mixing units, such as bp with percent Fix: Convert first: 1% = 100 bp. Then multiply.

Exam tips

• Expect conceptual contrast questions: VaR versus scenarios, and historical versus hypothetical.
• Read the wording on probability. Scenarios usually carry none unless stated.
• For calculations, check whether duration or spread duration must be multiplied by the shock.
• Link plausibility to consistent factor moves, which is the main theme of this chapter.
• Prefer answers that say tools complement each other over answers that call one superior.
• Read which factor is core and which is peripheral before computing. The question often swaps the roles.
• Check units and horizons first. Mismatched horizons are a common trap.
• Use sign logic to remove options quickly. The implied move has the same sign as ρ times the shock.