FRM Exam Part I · Modeling Non-Parallel Term Structure Shifts and Hedging

Multi-Factor Hedging with Swaps and Futures Using Key Rates and PCA

Updated 11 October 2026 · Fact-checked

Multi-factor hedging offsets a portfolio's exposure to several curve factors at once, such as key rates or principal components. Measure the portfolio's exposure to each factor, measure each hedge instrument's exposure to the same factors, then solve one equation per factor so the combined exposure is zero. You need as many instruments as factors.

Understand Multi-Factor Hedging with Swaps and Futures

A single duration hedge protects against a parallel shift only. Real curves twist and bend. A portfolio can have zero net DV01 and still lose money when short rates fall and long rates rise. Multi-factor hedging fixes this by controlling exposure to more than one factor.

The factors come in two main forms. Key rate factors move one maturity point (for example 2-year, 5-year, 10-year) and let neighbouring rates adjust by interpolation. Principal component factors are statistical shapes of curve moves, usually level, slope and curvature. Level is the largest and explains most of the variation.

For each factor, you compute the portfolio's exposure, such as a key rate 01 (the dollar change for a 1 bp move in that key rate) or a dollar change per unit of a principal component. You do the same for each hedge instrument: swaps, Treasury futures, or Eurodollar-type futures. Each instrument usually has exposure to several factors, not just one.

The hedge is a set of simultaneous linear equations. For each factor, portfolio exposure plus the sum of (position × instrument exposure) must equal zero. With two factors you need two instruments and solve a 2×2 system. With three factors you need three instruments.

The hedge is only as good as the factor model. It neutralises the chosen factors to first order. It leaves convexity, unmodelled factors and basis risk (for example, futures with a cheapest-to-deliver switch) unhedged. You must rebalance as exposures change.

Key formulas to remember

Key rate 01 of a position: KR01(k) = change in value for a 1 bp fall in key rate k, with other key rates unchanged; Sign convention matters. State whether exposure is per rate fall or per rate rise and keep it the same for the portfolio and the hedges.
Sum of key rate exposures: Σ KR01(k) ≈ DV01; Key rate exposures add up to the parallel-shift DV01. Use this as a quick check on your inputs.
Multi-factor hedge condition: E_P(j) + Σ N_i × E_i(j) = 0 for every factor j; E_P(j) is portfolio exposure to factor j, E_i(j) is exposure of one unit of instrument i, N_i is the position in instrument i. Solve for all N_i together.
Two-factor, two-instrument solution: N_B = −E_P(2) ÷ E_B(2) if instrument A has no exposure to factor 2; then N_A = −(E_P(1) + N_B × E_B(1)) ÷ E_A(1); Works when instruments are triangular, with A touching only factor 1. Otherwise solve the two equations by substitution.
Factor exposure of a portfolio: E_P(j) = Σ over holdings of (holding value × exposure of holding to factor j); Exposures are additive across positions, which is why the hedge equations are linear.

How to solve Multi-Factor Hedging with Swaps and Futures questions

Use this method for any multi-factor hedge question, whether it uses key rates or principal components.

1Identify the factors (for example 2-year and 10-year key rates, or level and slope) and the number of hedge instruments. They must match.
2Write down the portfolio's exposure to each factor, with its sign and units.
3Write down each instrument's exposure to each factor per contract or per unit of notional.
4Set up one equation per factor: portfolio exposure + Σ position × instrument exposure = 0.
5Solve the system. Start with the factor that only one instrument touches, if there is one, then back-substitute.
6Interpret signs: a negative futures position means selling contracts. A short futures position or a pay-fixed swap loses value when rates fall, so either one offsets a portfolio that gains when rates fall. Do not link a negative position to a swap side unless the question states its sign convention. Check how the instrument's exposure sign is defined in the question.
7Round to whole contracts at the end only, and check by plugging the numbers back into every equation.

Quickest way: Triangular solve and back-substitution

When to use it: Use when one instrument is clearly tied to the long end (or to one factor) and the other covers the short end, which is how most exam questions are built.

Pick the factor where only one instrument has material exposure (usually the longest key rate). Solve that instrument's position first.
Plug that position into the other factor's equation and solve for the second instrument.
Check signs: if the portfolio is long duration at both points, expect short hedges.
Confirm by adding the hedged exposures for each factor. Both must be zero.
If the options are far apart, estimate with the rounded numbers before doing full arithmetic.

Common mistakes in Multi-Factor Hedging with Swaps and Futures

Hedging with one instrument per portfolio DV01 and calling it done.
Single-factor duration hedging is the habit from earlier chapters.
Fix: Count the factors. If the question lists exposures at several key rates or principal components, you need one instrument per factor.
Ignoring an instrument's exposure to the other factors.
A 10-year futures contract is thought of as only a 10-year hedge, but it also has exposure at shorter key rates.
Fix: Include every nonzero exposure in each equation. Solve the longest-end instrument first, then adjust the short-end one.
Mixing sign conventions between the portfolio and hedges.
Some data are per 1 bp rise and others per 1 bp fall.
Fix: Convert everything to one convention before writing the equations.
Using the wrong equation sign, so the hedge doubles the exposure.
Students set the hedge exposure equal to the portfolio exposure instead of making the total zero.
Fix: Always write portfolio + hedge = 0. Then verify the hedged total numerically.
Believing a PCA hedge removes all curve risk.
Level, slope and curvature explain most, not all, of curve movement.
Fix: Say that residual risk remains from omitted factors, convexity and basis. Only the chosen factors are neutralised.
Rounding contract numbers in the middle of the solve.
Wanting tidy numbers early.
Fix: Keep full precision until the last step, then round to whole contracts.

Worked examples

Example 1

A bond portfolio gains $12,000 for a 1 bp fall in the 2-year key rate and $30,000 for a 1 bp fall in the 10-year key rate (KR01 = +$12,000 at the 2-year and +$30,000 at the 10-year, each the value change per 1 bp fall in that key rate with other key rates unchanged). You hedge with 2-year note futures (per contract: 2-year KR01 = $40, 10-year KR01 = 0) and 10-year note futures (per contract: 2-year KR01 = $5, 10-year KR01 = $80), on the same sign basis. How many contracts of each should you take?

Show the solution

Let N_A be the position in 2-year futures and N_B the position in 10-year futures. A positive number is long.
10-year equation: 30,000 + 0 × N_A + 80 × N_B = 0.
Solve: N_B = −30,000 ÷ 80 = −375, so short 375 of the 10-year futures.
2-year equation: 12,000 + 40 × N_A + 5 × N_B = 0.
Substitute: 12,000 + 40 N_A − 1,875 = 0, so 40 N_A = −10,125.
N_A = −10,125 ÷ 40 = −253.125, which rounds to short 253.
Check: 2-year: 12,000 − 10,120 − 1,875 = 5 (rounding only). 10-year: 30,000 − 30,000 = 0.

Answer: Short about 253 contracts of 2-year note futures and short 375 contracts of 10-year note futures.

Example 2

A portfolio has exposure of +$50,000 to the level factor and −$20,000 to the slope factor (dollar value change per one unit of each principal component shock). Instrument 1 has exposures of 100 to level and 40 to slope per contract. Instrument 2 has exposures of 80 to level and −60 to slope per contract. Find the positions that neutralise both exposures.

Show the solution

Let n1 and n2 be the positions in instruments 1 and 2.
Level: 50,000 + 100 n1 + 80 n2 = 0.
Slope: −20,000 + 40 n1 − 60 n2 = 0.
From the slope equation: 40 n1 = 20,000 + 60 n2, so n1 = 500 + 1.5 n2.
Substitute into the level equation: 50,000 + 100 × (500 + 1.5 n2) + 80 n2 = 0.
That is 50,000 + 50,000 + 150 n2 + 80 n2 = 0, so 100,000 + 230 n2 = 0 and n2 = −434.78.
n1 = 500 + 1.5 × (−434.78) = 500 − 652.17 = −152.17.
Check level: 50,000 − 15,217 − 34,783 = 0. Check slope: −20,000 − 6,087 + 26,087 = 0.

Answer: Short about 152 units of instrument 1 and short about 435 units of instrument 2.

Exam tips

Questions are usually 2×2 systems. Practise solving them in under two minutes, starting from the instrument with the simpler exposure pattern.
Read the sign convention in the stem (rise or fall, long or short) before writing any equation.
Check whether the stem gives exposures per contract or per $ notional. Convert to contracts at the end using contract size.
Expect conceptual options, such as which factor a hedge leaves unhedged or why PCA exposures need as many instruments as factors. The answer is usually residual risk from omitted factors or convexity.
Always plug your answer back into each factor equation. A quick check catches sign errors.

Practice questions from Modeling Non-Parallel Term Structure Shifts and Hedging

Multi-Factor Hedging with Swaps and Futures: frequently asked questions

How many hedging instruments do I need for a multi-factor hedge?

You need at least as many instruments as factors. With two key rates or two principal components, you need two instruments with different exposure patterns. If the instruments have similar exposures, the system is nearly unsolvable and the positions become very large.

What is the difference between a key rate hedge and a PCA hedge?

A key rate hedge neutralises the effect of moving specific maturity points, with neighbouring rates interpolated. A PCA hedge neutralises exposure to statistical shapes, usually level, slope and curvature, estimated from historical curve moves. The solving method is the same: one equation per factor.

Does a multi-factor hedge remove all interest rate risk?

No. It neutralises first-order exposure to the chosen factors only. Convexity, factors you left out, large non-linear moves and basis risk between the hedge and the portfolio remain, and the hedge needs rebalancing over time.

Why do the key rate exposures add up to DV01?

A parallel shift moves every key rate by the same amount. The total value change is the sum of the changes from each key rate move, so the key rate exposures add to the DV01 of the whole portfolio.