FRM Exam Part II · VaR Mapping

Principles and Purpose of VaR Mapping

Updated 11 October 2026 · Fact-checked

VaR mapping replaces each position in a portfolio with exposures to a small set of standard risk factors, such as key interest rates, equity indices or FX rates. You then compute VaR from the factor volatilities and correlations. It cuts data and computation, but adds approximation error.

Understand Principles and Purpose of VaR Mapping

A large bank holds thousands of positions: bonds with odd maturities, swaps, equities, options. Building a VaR model with a separate risk factor for every position is impractical. You would need a huge covariance matrix and clean price history for every instrument. Many instruments, like a new bond, have no history at all.

VaR mapping solves this. You choose a small set of primary risk factors (for example, a few zero-coupon rates, major equity indices, key FX rates). You then express each position as a combination of exposures to those factors. The portfolio becomes a vector of factor exposures, and VaR depends only on the factor volatilities and correlations.

Mapping is the step that defines the risk factors. Valuation is a separate choice that you make after mapping. You can use local valuation (for example, delta-normal) or full valuation of the mapped positions. Local valuation values the portfolio at the current point and uses sensitivities (exposures) to estimate changes. Full valuation revalues the mapped positions under each scenario. Full valuation is more accurate than local valuation, especially for options, but is slower. Local valuation is faster and more transparent, but it is only as good as the approximation.

Mapping brings trade-offs. Benefits: fewer parameters, a smaller and more stable covariance matrix, a way to handle instruments with no data, and easier aggregation across desks. Costs: basis risk or residual (specific) risk is dropped, linear mapping misses convexity and optionality, and a poor choice of factors can understate or misstate risk.

Good mapping preserves what matters: the market value of the position, its market risk (sensitivity to the factors), and ideally its cash-flow timing. A common rule in the cash-flow mapping approach is that the mapped positions must keep the same present value and the same risk as the original.

Key formulas to remember

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.

How to solve Principles and Purpose of VaR Mapping questions

Use this sequence for any mapping question, whether it asks for a concept, a calculation or a comparison of methods.

1Identify the positions and the instrument type: linear (bond, forward, swap, stock) or nonlinear (option).
2Decide the risk factors that drive each position, and check that data exist for those factors.
3Express each position as exposures to the factors. Make sure the mapped value equals the original value.
4Collect factor volatilities and correlations over the VaR horizon, and scale for horizon if needed.
5Compute portfolio VaR with z × √(xᵀΣx), or z × |x| × σ for one factor.
6Interpret the result: state the confidence level, horizon and what has been ignored (specific risk, convexity).
7If the question compares methods, keep mapping and valuation apart. Mapping to few factors needs less data and a smaller covariance matrix, but drops specific risk. Local (delta) valuation is fast but misses nonlinear risk. Full valuation is more accurate for nonlinear risks but slower.

Quickest way: Three-check shortcut for mapping questions

When to use it: Use it for conceptual MCQs on why and when to map, and for method comparisons.

Ask: does the option mention fewer risk factors or less data? That is a benefit of mapping.
Ask: does it mention ignored basis or specific risk, or missed nonlinear risk from delta (local) valuation? Those are costs of mapping to few factors and of linear treatment.
Ask: does it say local (delta) valuation is more accurate than full valuation for options? Reject it. Full valuation is the more accurate method, and it can be run on mapped positions.
For calculations, compute exposure × volatility × z first, then combine using correlation.

Common mistakes in Principles and Purpose of VaR Mapping

Saying local (delta) valuation is more accurate than full valuation.
Students link a simpler, cleaner model with a better one.
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.
Replacing a position with factors sounds like it removes idiosyncratic detail.
Fix: Mapping ignores specific risk and basis risk, so it can understate total risk, especially in concentrated portfolios.
Applying linear mapping to options without comment.
Delta mapping is easy, so students use it everywhere.
Fix: State that linear mapping misses gamma and convexity. For large moves, use delta-gamma or full valuation.
Mapping to factors that do not match the position's risk.
Students pick the nearest available index without checking fit.
Fix: Choose factors that capture the main drivers. A poor fit adds basis risk the model cannot see.
Forgetting to keep mapped value equal to original value.
Focus goes on risk and the value condition is dropped.
Fix: Check present value first. Then check that risk is preserved.
Mixing horizons or units when computing VaR from factors.
Volatility is quoted annually but the VaR horizon is daily.
Fix: Convert volatility to the VaR horizon before multiplying, for example σ(daily) = σ(annual) ÷ √252 where 252 trading days is assumed.

Worked examples

Example 1

A bank holds a USD 10 million equity portfolio mapped entirely to one index with beta 1.2. The index has daily volatility of 1.5%. Find the 1-day 99% parametric VaR using z = 2.326.

Show the solution

Exposure to the index = 1.2 × USD 10 million = USD 12 million.
Single-factor VaR = z × exposure × σ.
VaR = 2.326 × 12,000,000 × 0.015.
12,000,000 × 0.015 = 180,000.
180,000 × 2.326 = 418,680.

Answer: About USD 418,680 (roughly USD 0.42 million). This excludes the portfolio's specific risk, which the single-index mapping ignores.

Example 2

A risk manager maps a portfolio of 5,000 bonds and options to 20 risk factors. She must then choose between delta-normal (local) valuation and full valuation of the mapped positions. Which statement is correct? A) Delta-normal valuation is more accurate than full valuation for options. B) Mapping to 20 factors gives a smaller covariance matrix and handles instruments with no history, but specific risk is dropped and delta-normal valuation misses nonlinear risk. C) Full valuation needs less computation than delta-normal valuation. D) Mapping removes basis risk.

Show the solution

Test A: Delta-normal valuation approximates with sensitivities; full valuation is more accurate for options. A is wrong.
Test B: Fewer factors give a smaller covariance matrix. New instruments can be mapped to existing factors. Specific risk is dropped by mapping, and delta-normal valuation misses convexity. B is correct.
Test C: Full valuation revalues every position under each scenario and generally needs more computation. C is wrong.
Test D: Mapping introduces basis risk rather than removing it. D is wrong.

Answer: B

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.

Practice questions from VaR Mapping

Principles and Purpose of VaR Mapping: frequently asked questions

What is VaR mapping in FRM Part II?

It is the process of replacing a portfolio's positions with exposures to a smaller set of risk factors. VaR is then computed from those factors' volatilities and correlations. It makes VaR practical for large, complex portfolios.

Why use mapping in value at risk?

Mapping reduces the number of parameters to estimate and gives a smaller, more stable covariance matrix. It also lets you cover instruments with little or no price history by linking them to observable factors. The cost is some approximation error.

What is the difference between full valuation and local valuation in mapped VaR?

Mapping defines the risk factors. Valuation is a separate choice made afterwards. Full valuation revalues the positions under each scenario, so it captures nonlinear payoffs well but takes more computing. Local (delta) valuation uses factor exposures, so it is faster and simpler but can miss convexity. Mapping to few factors drops specific risk under either choice.

Does mapping remove idiosyncratic risk?

No. It ignores specific risk and basis risk between the position and the chosen factors. This can understate total risk, particularly for concentrated or poorly matched portfolios.