Show the reasoning behind VaR mapping and walk through the steps of the mapping procedure.

FRM® Part II · Market Risk Measurement and Management · VaR Mapping · FRM2.MR.5.a

The idea

A large portfolio holds thousands of instruments, many with short or no price histories, so estimating a covariance matrix across every position is impractical and noisy. VaR mapping solves this by replacing each position with exposures to a small set of primitive risk factors, such as equity indices, exchange rates and zero-coupon yields at standard maturities called vertices. The procedure has three steps. Value every position at current market prices, express each position as exposures to the chosen risk factors, and then compute portfolio VaR from the aggregated exposure vector and the factor covariance matrix. With z the normal quantile, x the exposure vector and Σ the factor covariance matrix, the portfolio VaR is VaR = z × √(x′Σx). A cash flow falling between two vertices is split between them so that both its market value and its market risk are preserved. Mapping trades some accuracy for tractability. The exam point is the purpose and the order of the steps: map first, then measure risk.

Formulas

Portfolio VaR from mapped exposures

VaR = z × √(x′ × Σ × x)

z: the standard normal quantile
x: the vector of exposures to the risk factors, x′ its transpose
Σ: the covariance matrix of the risk factors

Normal risk factors and linear exposures. Adding stand-alone factor VaRs instead ignores diversification.

Distinct entries in a covariance matrix

entries = n × (n + 1) ÷ 2

n: the number of assets or risk factors

Counts n variances plus n(n − 1) ÷ 2 covariances, since the matrix is symmetric. It grows with n², which is why positions are mapped.

A practice question

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

Why do risk managers map positions onto a limited set of primitive risk factors before computing portfolio VaR?

It removes position-specific risk entirely, so the VaR that results is a pure market-wide figure.Why not: Mapping simplifies the inputs; it does not delete risk, and specific risk drops out only if the mapping chooses to ignore it.
It cuts the number of volatilities and correlations that must be estimated and copes with instruments that have little price history. - correct
It guarantees a conservative VaR because every exposure is rounded up to the nearest vertex.Why not: Invents a rounding-up rule; mapping preserves value and risk rather than biasing VaR upward.
It removes the need to estimate volatilities, because each risk factor is assumed to have unit volatility.Why not: Mapping still needs factor volatilities and correlations; it only shrinks the matrix that has to be estimated.

Mapping replaces thousands of positions with exposures to a few risk factors, so the covariance matrix is small and well estimated, and new or illiquid instruments can be handled through their factor exposures. It does not eliminate any inputs; it reduces their number.

A harder one

Which sequence correctly describes the VaR mapping procedure?

Mark positions to market, express each position as exposures to the chosen risk factors, then compute portfolio VaR from the exposures and the factor covariance matrix. - correct
Estimate a covariance matrix for every individual instrument, then map the resulting VaR to the nearest risk factor.Why not: Reverses the order; mapping comes before risk measurement, precisely to avoid instrument-level covariance matrices.
Choose the risk factors, compute each factor's VaR, then add the factor VaRs to obtain the portfolio VaR.Why not: Adding factor VaRs ignores correlations and gives the undiversified figure, not the covariance-based VaR.
Mark positions to market, compute each position's VaR, then map the largest VaRs onto risk factors.Why not: Maps after risk has been computed; mapping must decompose positions into factor exposures first.

The three steps are valuation, mapping to factor exposures, and then VaR from x and the factor covariance matrix. Summing separate VaRs would ignore diversification, and building instrument-level covariances defeats the purpose of mapping.

The hardest tier

A portfolio is mapped to two risk factors with long exposures of USD 10 million and USD 6 million. Daily volatilities are 1.0 percent and 1.5 percent and the correlation is 0.3. Using the normal approximation, the one-day 95 percent VaR is closest to:

USD 313,000Why not: Adds the two stand-alone VaRs, which assumes a correlation of 1.
USD 221,000Why not: Sets the correlation to zero, treating the factors as independent.
USD 185,000Why not: Uses a correlation of -0.3, flipping the sign of the covariance term.
USD 252,000 - correct

Stand-alone amounts are 10m x 1.0% = 0.100m and 6m x 1.5% = 0.090m. Variance = 0.100² + 0.090² + 2 x 0.3 x 0.100 x 0.090 = 0.0235, so sigma = 0.1533m and VaR = 1.645 x 0.1533m = 0.252m. The undiversified sum, 313,000, is the ceiling.

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