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FRM Part II · FRM Exam Part II

VaR Mapping for FRM Part II: Chapter Guide

VaR mapping replaces each position in a portfolio with exposures to a small set of common risk factors, such as zero-coupon bond prices, FX rates, equity indices or betas. You then compute VaR from the factor volatilities and correlations. To solve questions, identify the factors, find the exposure to each, then aggregate the risk.

What this chapter covers

VaR mapping solves a practical problem. A large bank holds thousands of positions: bonds, swaps, shares, options. Modelling every one with its own return history is not workable. Mapping reduces each position to exposures on a limited set of risk factors. Risk is then measured on those factors, using their volatilities and correlations.

The chapter moves from simple to harder cases. Fixed-income mapping assigns cash flows to vertices on the zero-coupon curve. Forwards, FRAs and swaps are then treated as combinations of long and short zero-coupon positions. Equity portfolios are mapped to market indices through beta. Options are nonlinear, so they need a delta (and sometimes gamma) approximation, which is only valid for small moves.

This chapter links to the rest of Part II. It supports the market risk topic, where VaR, expected shortfall and model limits are tested. It also feeds stress testing and risk aggregation. If you understand mapping, you can judge why a VaR number may be wrong, which is the kind of interpretation Part II questions ask for.

VaR mapping is a compact chapter, but it is a source of numerical questions that reward method over memory. The same steps recur across instruments: pick the risk factors, compute the exposure, apply volatility and correlation. Once you hold that pattern, you can answer calculation questions quickly and also handle conceptual ones on mapping error, the limits of delta approximations and the choice of vertices. Given 80 questions in 4 hours, the speed you gain here helps you across the whole market risk section.

VaR Mapping: topics in the order to study them

  1. 1Principles and Purpose of VaR MappingIt defines risk factors, exposures and the trade-off between accuracy and simplicity, which every later topic uses.
  2. 2Mapping Fixed-Income PortfoliosCash-flow mapping to zero-coupon vertices is the base method; derivatives mapping reuses it directly.
  3. 3Mapping Linear Derivatives: Forwards, FRAs and SwapsThese are built from long and short zero-coupon positions, so they follow once you can map bond cash flows.
  4. 4Mapping Equity Portfolios and Beta MappingThis is a separate, simpler idea based on index exposure and beta, best learned once the factor logic is familiar.
  5. 5Mapping Nonlinear Instruments and OptionsIt is the hardest part, because delta and gamma approximations extend the linear mapping and have clear limits.

How to prepare VaR Mapping

Treat this chapter as one method applied to different instruments. Build the method first, then practise it on each instrument type.

  1. Write the three-step method on one line: choose factors, compute exposures, aggregate with volatilities and correlations.
  2. Learn why mapping is used: fewer risk factors, faster computation, and usable data for instruments with little history.
  3. Practise mapping a bond's cash flows to vertices, and check that the mapping preserves present value and risk.
  4. Rewrite a forward, FRA and swap as long and short zero-coupon positions, and do it until it feels routine.
  5. Do equity examples with beta: exposure to the index equals position value times beta, and the rest is specific risk.
  6. For options, compute the delta-equivalent position, then note when gamma matters and why the linear approximation breaks for large moves.
  7. Finish with mixed timed questions, and for each one state the measure, the method and what the result means.

Common mistakes in VaR Mapping

  • Adding the VaR of individual positions to get portfolio VaR

    Fix: Aggregate mapped exposures using volatilities and correlations. Simple addition only applies with perfect correlation.

  • Mapping a cash flow to a single vertex without preserving value and risk

    Fix: Split the cash flow between the two adjacent vertices so that present value is kept and the risk matches, then check the totals.

  • Treating a swap or FRA as one instrument rather than a set of zero-coupon positions

    Fix: Write out every cash flow, assign signs for long and short legs, and map each to the curve.

  • Using beta mapping and assuming all risk is captured

    Fix: State that beta covers only systematic risk. Specific risk remains, and it is larger for concentrated portfolios.

  • Applying delta to options for large price moves

    Fix: Use delta only for small moves. Mention gamma, or full revaluation, when the move is large or the option is near the money close to expiry.

  • Mixing up position value and exposure

    Fix: Exposure is the amount sensitive to a factor, such as value times beta or delta times underlying. Calculate it before applying any volatility.

Last-day revision: VaR Mapping

  • 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.

VaR Mapping practice questions

VaR Mapping in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

VaR Mapping: frequently asked questions

What is VaR mapping in simple terms?

It means expressing every position as exposure to a few standard risk factors, such as zero-coupon rates, FX rates or equity indices. VaR is then computed on those factors. This keeps the calculation manageable for large portfolios.

Why are forwards and swaps mapped to zero-coupon bonds?

Their cash flows can be rebuilt from long and short zero-coupon positions at different maturities. That lets you use the same factor set as for bonds. It also makes the risk easy to aggregate.

Is delta mapping accurate for options?

Only for small moves in the underlying. Options are nonlinear, so for large moves delta understates or misstates the change in value. Gamma or full revaluation improves the estimate.

How should I practise this chapter?

Start with one worked example per instrument type, then do mixed questions under time limits. For each one, name the risk factors, compute the exposures and explain the result in words. That mirrors the applied style of Part II questions.