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FRM Exam Part II · VaR Mapping

Mapping Forwards, FRAs and Swaps for VaR

Updated 11 October 2026 · Fact-checked

Mapping replaces a linear derivative with a portfolio of simple building blocks whose risk you can measure. A forward is a long position in one zero-coupon bond and a short position in another, or a spot asset plus a bond. An FRA is long one zero and short another. A swap is a fixed-rate bond against a floating-rate bond.

Understand Mapping Linear Derivatives: Forwards, FRAs and Swaps

VaR needs risk factors with known volatilities and correlations, such as spot prices and zero-coupon bond prices at standard maturities. A forward, FRA or swap is not one of these. So you map it: you rewrite the instrument as a combination of positions in the basic risk factors, then compute VaR on that combination.

Start with a forward to buy an asset at price K at time T. At expiry you pay K and receive the asset. Today that equals a long position in the spot asset and a short position in a zero-coupon bond that pays K at T. For a currency forward, the long leg is the foreign zero-coupon bond converted at spot into your home currency, and the short leg is the home-currency zero. So you carry three risk factors: the spot FX rate, the foreign bond price and the domestic bond price.

An FRA fixes a rate for a future period from T1 to T2. The buyer of the FRA, who pays the fixed rate and receives the floating rate, is economically long a zero maturing at T1 and short a zero maturing at T2, with the notional received at T1 and repaid, with the fixed rate, at T2. This is the buyer's position, equivalent to borrowing the notional over the period. The two legs have opposite signs and similar values, so the position is a spread. Its risk comes from the difference in the two zero-rate moves, not their levels.

A swap is a series of such cash flows. A pay-fixed, receive-floating swap is a long position in a floating-rate note and a short position in a fixed-coupon bond. The floating-rate note resets to par at each reset date, so it behaves like a zero maturing at the next reset. The fixed leg is a strip of zeros, one for each coupon date. In practice you map each cash flow to the standard vertices, usually by splitting its present value between the two adjacent vertices.

After mapping, each position has a present value (the exposure) on each risk factor. VaR then follows the usual delta-normal method: exposures, volatilities, correlations, then a z-value. Mapping is exact for the cash-flow decomposition. The approximation comes from the linear treatment of price changes and the allocation to vertices.

Key formulas to remember

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.

How to solve Mapping Linear Derivatives: Forwards, FRAs and Swaps questions

Use the same sequence for any forward, FRA or swap mapping question.

  1. 1Identify the instrument and your side: long or short, pay fixed or receive fixed, buy or sell forward.
  2. 2List the cash flows and their dates. Forward: one outflow K at T and the asset. FRA: notional at T1 and T2. Swap: each fixed coupon and the floating leg.
  3. 3Convert each cash flow into a zero-coupon bond position. Give each a sign: positive for cash you receive, negative for cash you pay.
  4. 4Compute the present value of each leg using the given discount factors or zero rates. These PVs are the exposures.
  5. 5If a cash flow falls between vertices, split its PV across the two nearest vertices, keeping value (and risk) matched as the question instructs.
  6. 6Attach the volatility of each risk factor and the correlations between them.
  7. 7Compute portfolio volatility with the exposures, then multiply by the z-value and by the horizon scaling √t if needed.
  8. 8Check the result: a long-short spread should have lower VaR than the sum of the legs' stand-alone VaRs.

Quickest way: Leg-by-leg VaR with a correlation shortcut

When to use it: Use when the question gives two risk factors with a correlation, as in FRAs, short-dated swaps and forwards with two legs.

  1. Write the PV of each leg with its sign.
  2. Compute each leg's stand-alone VaR as an absolute amount: |PV| × z × σ.
  3. For a long-short pair, combine the absolute amounts using VaR_p = √(VaR₁² + VaR₂² − 2ρ × VaR₁ × VaR₂). The minus sign in the cross term comes from the opposite signs of the legs. If both legs were long, the cross term would be plus.
  4. If ρ = 1 the VaR is the difference of the legs. If ρ = 0 it is the root of the sum of squares. Use these to eliminate wrong options quickly.

Common mistakes in Mapping Linear Derivatives: Forwards, FRAs and Swaps

  • Adding the stand-alone VaRs of the two legs

    You treat the legs as independent assets, forgetting that one is short.

    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

    Notional is the number in the question, so it feels like the position size.

    Fix: Always discount each cash flow. VaR exposures are the PVs of the mapped zero positions.

  • Forgetting the foreign zero's conversion to home currency in a currency forward

    You model only the FX spot and one bond.

    Fix: Map into three factors: spot FX, the foreign zero converted at spot, and the domestic zero. Each has its own volatility.

  • Mapping the floating leg of a swap as a full strip of zeros

    You copy the treatment of the fixed leg.

    Fix: The floating leg is worth par at the next reset, so map it to one zero at the next reset date.

  • Getting the signs of an FRA wrong

    Mixing up buyer and seller, and fixed and floating payer.

    Fix: The FRA buyer pays fixed and receives floating, so is long the T1 zero and short the T2 zero. Check the direction with a rate-rise thought experiment: the buyer gains.

  • Scaling a one-day VaR by the number of days instead of its square root

    Rushing the final step, or assuming risk grows in a straight line with time.

    Fix: Multiply by √(horizon in days), not by the number of days. Apply the z-value once.

Worked examples

Example 1

A bank is long a 1-year forward to buy EUR 1,000,000 at a fixed price of USD 1.10 per EUR. Spot is 1.08. The 1-year EUR zero price is 0.97 and the 1-year USD zero price is 0.95. Map the forward and give its present value in each risk factor.

Show the solution
  1. The forward is long EUR zero and short USD zero. Because the EUR zero is converted into USD at spot, there are three risk factors: EUR/USD spot, the 1-year EUR zero and the 1-year USD zero.
  2. Long leg: EUR 1,000,000 × 0.97 = EUR 970,000. In USD at spot 1.08 this is 970,000 × 1.08 = USD 1,047,600.
  3. Short leg: you pay USD 1,100,000 at T. PV = 1,100,000 × 0.95 = USD 1,045,000, held as a short position.
  4. Mapping: long USD 1,047,600, which is exposed to both EUR/USD spot and the EUR zero price, and short USD 1,045,000 in the USD zero.
  5. Forward value = 1,047,600 − 1,045,000 = USD 2,600.

Answer: Three risk factors: EUR/USD spot, the EUR 1-year zero and the USD 1-year zero. The long position of USD 1,047,600 is exposed to both spot FX and the EUR zero price. The short position of USD 1,045,000 is in the USD 1-year zero. The forward is worth USD 2,600.

Example 2

A bank buys an FRA on USD 10,000,000. Mapped, it is long a zero with PV USD 9,800,000 and short a zero with PV USD 9,700,000. Daily volatilities of the two zeros are 0.10% and 0.12% and the correlation is 0.9. Find the 1-day 95% VaR (z = 1.645) to the nearest USD.

Show the solution
  1. Leg 1 dollar volatility: 9,800,000 × 0.0010 = 9,800.
  2. Leg 2 dollar volatility: 9,700,000 × 0.0012 = 11,640, with a negative sign.
  3. Variance = 9,800² + 11,640² − 2 × 0.9 × 9,800 × 11,640.
  4. 9,800² = 96,040,000. 11,640² = 135,489,600.
  5. Cross term = 2 × 0.9 × 9,800 × 11,640 = 1.8 × 114,072,000 = 205,329,600.
  6. Variance = 96,040,000 + 135,489,600 − 205,329,600 = 26,200,000.
  7. Standard deviation = √26,200,000 ≈ 5,118.6.
  8. VaR = 1.645 × 5,118.6 ≈ 8,420.

Answer: About USD 8,420. This is far below the sum of the stand-alone VaRs because the two legs are highly correlated and opposite in sign.

Exam tips

  • Draw the two legs with signs before any arithmetic. Most wrong answers come from a sign error.
  • If an option shows the simple sum of the leg VaRs, it is usually the trap. The correct answer is lower when ρ is high and the legs are opposite.
  • Know the pay-fixed swap = long floating note, short fixed bond rule and its reverse. Questions often flip the side.
  • Remember that a forward on a currency has three risk factors, and say which one is missing if the question gives only two.
  • Check units: use PV in the currency of VaR, and convert foreign legs at spot.

Practice questions from VaR Mapping

Mapping Linear Derivatives: Forwards, FRAs and Swaps: frequently asked questions

Why do we map a forward to a zero-coupon bond?

Zero-coupon bond prices at standard maturities are the risk factors with data on volatility and correlation. Mapping turns a contract with one future payment into those factors so VaR can be computed.

How is an FRA mapped for VaR?

The buyer of an FRA is long a zero maturing at the start of the period and short a zero maturing at the end. It is a spread position, so its risk depends on the correlation between the two zero rates.

How do you map an interest rate swap?

A pay-fixed swap is a long floating-rate note and a short fixed-coupon bond. The floating note is mapped to a zero at the next reset. The fixed bond is mapped to a zero at each coupon date.

Is mapping exact?

The cash-flow decomposition is exact for these linear instruments. The approximations are the linear delta-normal treatment of price changes and the allocation of cash flows to standard vertices.