VaR Mapping: formula sheet

Key formulas

Linear mapping of a position

Position value ≈ Σ (xᵢ × Fᵢ), where xᵢ is the exposure to factor i and Fᵢ is the factor

Exposure xᵢ is the delta or sensitivity of the position to factor i. This is valid for small moves and linear instruments.

Portfolio VaR from mapped exposures (normal)

VaR = z × √(xᵀ Σ x), where Σ is the covariance matrix of factor returns over the horizon

x is the vector of money exposures to each factor. z is the normal quantile (1.645 at 95%, 2.326 at 99%).

Single-factor VaR

VaR = z × |x| × σ

σ is the factor's return volatility over the VaR horizon, in the same units as x.

Horizon scaling of volatility

σ(T days) = σ(1 day) × √T

Valid under independent, identically distributed returns with zero drift assumptions.

Mapping conditions

Mapped value = original value; mapped risk = original risk

Cash-flow mapping splits a cash flow between two adjacent vertices so both conditions hold.

Present value of a cash flow

PV = CF ÷ (1 + y)^t

Use the zero (spot) rate y for that date t, with the compounding convention in the question.

Value-preserving split

PV = V_a + V_b, with V_a = α × PV and V_b = (1 − α) × PV

V_a sits at the shorter vertex and V_b at the longer vertex.

Interpolated volatility

σ_cf = σ_a + (t − t_a) ÷ (t_b − t_a) × (σ_b − σ_a)

Linear interpolation of vertex price volatilities, where t_a < t < t_b.

Risk-preserving condition

σ_cf² = α² σ_a² + (1 − α)² σ_b² + 2 α (1 − α) ρ σ_a σ_b

Solve this quadratic for α. Take the root between 0 and 1.

Duration mapping vertex

Map the bond's PV to the vertex at its (Macaulay) duration

If duration lies between vertices, split between the two, or use the nearest one, as the question states.

Portfolio VaR from vertex positions

VaR = z × √(x′ Σ x) , with x = vector of vertex PVs

Σ is the covariance matrix of vertex price changes. Individual vertex VaR = z × σ × PV.

Forward as spot plus bond

Long forward = long spot asset S + short zero-coupon bond paying K at T

Value today = S − K × P(0,T) for a non-dividend asset, where P is the zero price.

Currency forward mapping

Long foreign-currency forward = long foreign zero (in home value) + short domestic zero

Foreign zero's home value = S × foreign discount factor × notional. Three factors: spot FX, foreign zero, domestic zero.

FRA mapping

Buyer of FRA (T1 to T2) = long zero at T1 + short zero at T2

Buyer pays fixed, receives floating. Present values of the two legs are PV of notional at T1 and PV of notional × (1 + K × τ) at T2.

Swap mapping

Pay-fixed swap = long floating-rate note − fixed-coupon bond

Floating note is worth par just after reset, so treat it as a zero to the next reset date.

Delta-normal VaR of mapped portfolio

VaR = z × √(xᵀ Σ x), where x = vector of PV exposures and Σ = covariance matrix of risk-factor returns

Use the z-value for the confidence level, e.g. 1.645 at 95% and 2.326 at 99% one-tailed.

Two-position VaR

σ_p = √(σ₁²x₁² + σ₂²x₂² + 2ρσ₁σ₂x₁x₂)

For long-short mappings x₂ is negative, so the correlation term reduces risk when ρ is high.

Single-index model

R_i = α_i + β_i × R_M + ε_i

ε_i is uncorrelated with R_M and with other stocks' residuals.

Beta

β_i = Cov(R_i, R_M) ÷ Var(R_M) = ρ_iM × σ_i ÷ σ_M

Measures sensitivity to the index, not total risk.

Portfolio beta exposure

Index-equivalent exposure = Σ β_i × W_i

Use value weights. Short positions have negative W_i.

Systematic (mapped) VaR

VaR_sys = z × σ_M × Σ β_i × W_i

σ_M must be for the same horizon as the VaR. Take the absolute value of the exposure.

Variance decomposition

σ_i² = β_i² × σ_M² + σ_ε,i²

Total variance equals systematic plus specific.

Portfolio specific risk

σ_ε,p = √(Σ W_i² × σ_ε,i²)

Uses squared position values because residuals are uncorrelated.

Total VaR

VaR_total = z × √[(σ_M × Σ β_i W_i)² + Σ W_i² σ_ε,i²]

Combine systematic and specific by squares. Do not add the two VaRs.

Equal-position diversification

σ_ε,p = σ_ε × W_total ÷ √N

For N equal positions with equal residual volatility σ_ε.

Horizon scaling

σ_T = σ_daily × √T

Assumes independent, identically distributed returns.

Delta approximation

ΔV ≈ Δ × ΔS

Δ = ∂V/∂S. Valid for small moves in S. Treats the option as a linear position.

Delta-normal option VaR

VaR(option) ≈ |Δ| × S × σ × z × √t

σ is volatility of returns for the horizon unit, z is the normal quantile (1.645 at 95%, 2.326 at 99%). Use the absolute value because VaR is a loss amount.

Delta-gamma approximation

ΔV ≈ Δ × ΔS + ½ × Γ × (ΔS)²

Γ = ∂²V/∂S². Second-order Taylor expansion of option value.

Worst-case S move at the VaR quantile

ΔS* = ± S × z × σ × √t

Evaluate delta-gamma at the adverse move. For a long call or put (Γ > 0) check both ends; the loss is usually at the end the delta points to.

Delta-gamma loss at the adverse move

Loss ≈ −(Δ × ΔS* + ½ × Γ × (ΔS*)²)

This quantile shortcut is exact only when the loss is monotonic in S. It is an approximation otherwise.

Delta of a position

Position delta = number of options × Δ × contract size

Convert to underlying-equivalent exposure before applying VaR.

Quick revision

• Mapping replaces positions with exposures to a small set of common risk factors.
• Main benefits: fewer inputs, faster computation, and coverage of instruments with thin data.
• Mapping adds approximation error, so VaR depends on the factors and vertices you choose.
• Fixed-income mapping assigns each cash flow to nearby zero-coupon vertices.
• A good cash-flow map preserves the present value and the risk of the original cash flow.
• Forwards, FRAs and swaps decompose into long and short zero-coupon bond positions.
• A swap can be seen as a fixed-rate bond against a floating-rate bond.
• Equity exposure to the index is position value times beta.
• Beta mapping leaves out specific risk, so it understates risk in undiversified portfolios.
• Option delta-equivalent exposure is delta times the underlying position value.
• The delta approximation holds for small moves; gamma matters for large moves.
• Portfolio VaR from mapped factors uses factor volatilities and correlations, not simple addition.

Common mistakes

• Saying local (delta) valuation is more accurate than full valuation. Fix: Remember local valuation trades accuracy for speed and tractability. Full valuation is the benchmark for accuracy, and it can be run on mapped positions.
• Treating mapping as a way to remove all risk of a position. Fix: Mapping ignores specific risk and basis risk, so it can understate total risk, especially in concentrated portfolios.
• Mapping a coupon bond to its maturity vertex and calling it exact. Fix: Remember it ignores coupons. It typically overstates the bond's risk when vertex volatility rises with maturity, because the true average timing is shorter than maturity.
• Using the bond's maturity instead of its duration in duration mapping. Fix: Duration is the weighted average time of cash flows. Use it, not maturity, to pick the vertex.
• Adding the stand-alone VaRs of the two legs Fix: Use signed exposures and the correlation. Adding gives an upper bound only when both are the same sign and ρ = 1.
• Using face value instead of present value as the exposure Fix: Always discount each cash flow. VaR exposures are the PVs of the mapped zero positions.
• Using total stock volatility instead of index volatility in the mapped VaR. Fix: After mapping, the position is an index position. Use σ_M only; stock-specific volatility belongs in the residual term.
• Adding systematic VaR and specific VaR directly. Fix: Residual risk is uncorrelated with the market, so combine variances. Total = √(sys² + spec²).
• Applying delta-normal VaR to a short option position and treating it as safe. Fix: Check gamma. A short gamma position can lose far more than delta suggests in large moves.
• Forgetting to multiply by contract size and number of options. Fix: Convert to position delta first, then to a currency exposure.

Exam tips

• Expect conceptual MCQs: state the purpose (fewer factors, handle missing data) and the cost (approximation, missing nonlinear and specific risk).
• In comparisons, keep the two choices apart. Mapping defines the risk factors and cuts data needs. Valuation is then local (faster, approximate) or full (more accurate, slower).
• For calculations, check units and horizon first. Most lost marks come from mixing daily and annual volatility.
• Look for the phrase 'preserve value and risk' when a question asks what a good mapping must do.
• Link this topic to later mapping chapters: fixed income, forwards and swaps, equities and options each use the same principle.
• Know the ranking: principal mapping is least accurate, cash-flow mapping is most accurate, and duration mapping sits between.
• Check that mapped PVs add up to the bond PV. It catches most arithmetic slips quickly.
• Questions often give ρ = 1 to simplify α. Look for it before building the quadratic.