Mapping Positions to Risk Factors
Risk-factor mapping replaces a list of instruments with exposures to common drivers such as yield-curve tenors, FX rates, spreads and volatility. Preserve the important sensitivities before aggregating risk or designing a hedge.
The method at a glance
Illustrative signed loss sensitivities, currency per bp.
| Position | 2-year | 10-year |
|---|---|---|
| Bond | +1,000 | +4,000 |
| Hedge | 0 | -5,000 |
| Net | +1,000 | -1,000 |
| Twist | +10 bp | -10 bp |
Twist loss: 1,000 × 10 + (-1,000) × (-10) = 20,000.
| Concept | How it works | Watch for |
|---|---|---|
| Choose factors | Map the price drivers, not merely the asset labels. | Putting every bond into one parallel-rate bucket. |
| Measure exposure | Keep the units and signs of DV01, delta and vega consistent. | Adding exposures quoted in different currencies or units. |
| Check residuals | Test curve twists, basis moves and nonlinear effects. | Equating zero net parallel DV01 with zero interest-rate risk. |
01Choose factors
How it works: Map the price drivers, not merely the asset labels.
Watch for: Putting every bond into one parallel-rate bucket.
02Measure exposure
How it works: Keep the units and signs of DV01, delta and vega consistent.
Watch for: Adding exposures quoted in different currencies or units.
03Check residuals
How it works: Test curve twists, basis moves and nonlinear effects.
Watch for: Equating zero net parallel DV01 with zero interest-rate risk.
Original worked example
A hedge that misses a curve twist
A bond has signed loss sensitivity of +1,000 per basis-point rise at the two-year tenor and +4,000 at ten years. A hedge contributes -5,000 at ten years.
- 1
Net parallel DV01 = 1,000 + 4,000 - 5,000 = 0.
- 2
For a twist of +10 basis points at two years and -10 at ten years, compute each bucket.
- 3
Loss ≈ 1,000 × 10 + (-1,000) × (-10) = 20,000.
Parallel exposure is zero, but the twist produces a 20,000 first-order loss. Tenor mapping reveals what the aggregate hid.
Illustrative assumptions and calculations for learning. Not observed market data or an official exam item.
Apply the method and check its limits
Before you start: Signed sensitivities, basis points and yield-curve tenors.
Replace instrument labels with common drivers
Approximate loss = Σ signed loss sensitivityᵢ × factor shockᵢ
Map bonds to curve tenors and credit spreads, foreign assets to exchange rates, and options to underlying price and volatility. A position can depend on several factors. Sensitivities must share currency, units, horizon and sign conventions before aggregation. A risk-free rate hedge does not necessarily hedge a corporate bond’s credit spread.
Test the hedge against more than a parallel shift
In the example, the hedge offsets the sum of the two-year and ten-year sensitivities. After the hedge, the tenor exposures are +1,000 and -1,000 per bp. They cancel for a parallel shift but not for a curve twist. Applying +10 bp to the two-year point and -10 bp to the ten-year point produces an approximate loss of 20,000 under this loss-sensitivity convention.
A proxy creates basis risk
Using an index to represent a specific stock or a benchmark yield to represent a credit instrument leaves residual exposure. Validate the proxy under ordinary and stressed conditions. For options, include volatility and curvature where material; a linear factor map may otherwise understate losses after a large move.
Work through the topic
Convert a portfolio into measurable factor exposures, reconcile the mapped approximation with valuation, and recognize what netting across factors can conceal.
Build the position-to-factor inventory
Start with contractual features: currency, maturity, payment dates, reference rates, credit terms and embedded options. Then identify the market variables that change value. A floating-rate note, fixed-rate bond and interest-rate option may reference the same market while having different sensitivities. Mapping by an instrument label alone can therefore misrepresent the economic exposure.
Specify each factor’s units. An interest-rate change may be expressed in basis points, an equity return as a decimal percentage change and an exchange rate as units of one currency per another. Sensitivities must match those definitions. A currency quote inversion changes the interpretation of a move. A robust mapping record includes the valuation currency, sign convention and date of the sensitivity calculation.
For a simple foreign equity position, value in the home currency depends on both the local share price and the exchange rate. Its exact home-currency return is (1 + local return) × (1 + currency return) - 1 under a consistent quote. If local shares rise 5% and the foreign currency falls 3% against home currency, the combined return is 1.05 × 0.97 - 1 = 1.85%. Adding 5% and -3% gives a close first-order estimate of 2%, omitting the interaction.
Keep yield-curve tenors visible after aggregation
An aggregate duration summarizes exposure to a specified parallel yield move. It can hide offsetting sensitivities at different maturities. Suppose a portfolio has signed key-rate DV01 exposures of +10,000 at two years and -10,000 at ten years. Here positive DV01 means the value decreases by that amount for a one-basis-point increase in the corresponding rate, holding the other modeled nodes fixed.
The aggregate signed DV01 is zero. A parallel 5 bp increase gives a first-order change of -(10,000 × 5 + (-10,000) × 5) = 0. But a curve twist with the two-year rate up 5 bp and the ten-year rate down 5 bp gives -(10,000 × 5 + (-10,000) × (-5)) = -100,000. The portfolio is neutral to the stated parallel move, not neutral to all rate movements.
The exact result depends on how the curve is constructed and how shocks are interpolated between nodes. A key-rate system is a modeling choice, so document its tenors and bump method. Reconcile mapped sensitivities with full repricing for representative shocks. Large differences may reveal nonlinearity, missing factors or inconsistent pricing conventions rather than an arithmetic mistake in aggregation.
| Scenario | Two-year and ten-year shocks | Approximate value change |
|---|---|---|
| Parallel increase | +5 bp; +5 bp | 0 |
| Curve twist | +5 bp; -5 bp | -100,000 |
| Opposite twist | -5 bp; +5 bp | +100,000 |
Original curve example: identical aggregate DV01, different scenario results
- Parallel increase
- Two-year and ten-year shocks: +5 bp; +5 bp
- Approximate value change: 0
- Curve twist
- Two-year and ten-year shocks: +5 bp; -5 bp
- Approximate value change: -100,000
- Opposite twist
- Two-year and ten-year shocks: -5 bp; +5 bp
- Approximate value change: +100,000
Measure residual risk and refresh the map
A proxy maps an instrument to a related observable factor when direct data are unavailable or impractical. That creates basis risk: the instrument and proxy need not move together exactly. A regional commodity contract mapped to a global benchmark can retain location, quality and delivery differences. Historical closeness is evidence about a sample, not a contractual guarantee of future tracking.
Use scenario reconciliation to inspect the residual. Compare a fully repriced portfolio change with the change predicted by the mapped factors. Explain the difference using omitted factors, nonlinear terms, stale positions or pricing discrepancies. A residual that grows in stressed conditions deserves particular attention, since a map calibrated only to quiet periods may understate precisely the risk it is intended to describe.
Refresh mapping when positions, product features or sensitivities change. An option’s delta can shift as its underlying price and time to expiry change; a bond can approach a payment or exercise date. Store enough information to reproduce a reported exposure. The mapping output then feeds VaR, stress testing or hedging, but the statistical assumptions of those downstream methods remain separate choices.
Does a zero net factor exposure imply zero risk?
Only for the modeled first-order change in that factor under the stated assumptions. Other factors, nonlinear terms, basis differences and changes in sensitivities can remain. Inspect both gross and net exposures and test scenarios that challenge the apparent offset.
How is risk-factor mapping different from VaR?
Mapping describes what the portfolio is exposed to and how value responds. VaR combines a mapped or fully valued portfolio with a modeled distribution of changes to estimate a loss quantile. Good mapping is necessary for many VaR approaches, but it does not choose the distribution.
Try a changed assumption
For post-hedge sensitivities +1,000 and -1,000 per bp, what happens if both tenors rise 10 bp?
Show answer and reasoning
The first-order losses sum to +10,000 - 10,000 = 0. This is neutrality to the specified parallel move, not to every possible curve change.
Original learning check. No email or account needed.
Can exposures simply be netted?
Only after aligning factor definitions, units, currencies and valuation conventions. Even then, nonlinear, basis and liquidity risks may remain.
Sources and scope
Links checked 2026-10-10. GARP establishes exam scope; the technical references support the methods identified below. These are selected study notes and original examples, not complete curriculum coverage or official exam questions.
- GARP FRM study materials and 2026 learning objectives
Exam-domain scope and the official route to the annual learning objectives. This page does not certify coverage of every reading.
- MIT OpenCourseWare: fixed-income securities, pp. 19-20
Present-value pricing, modified duration and the convexity approximation.
- MIT: Factor Modeling, lecture 15
Factor exposures and covariance-based portfolio risk decomposition.
Prepared by MiloPrep editorial team. MiloPrep is independent of GARP. References do not imply endorsement or independent review of these notes.