FRM Part I · FRM Exam Part I

Modeling Non-Parallel Term Structure Shifts and Hedging

Non-parallel shifts mean different maturities on the yield curve move by different amounts. You model them with key rate shifts or principal components, measure exposure to each factor, then choose swap or futures positions so each factor exposure of the portfolio nets to zero. You solve it with simultaneous equations.

What this chapter covers

Duration and DV01 assume every rate moves by the same amount. Real curves steepen, flatten and twist. This chapter shows how to describe those moves and how to hedge against them.

You start by reviewing duration, DV01 and convexity, which measure price sensitivity to a parallel shift. Then you split the curve into key rates, measuring the price change when one maturity point moves and the others stay fixed. Next you meet principal components analysis (PCA), which finds a few statistical factors, usually called level, slope and curvature, that explain most curve movement. Finally you hedge: pick instruments such as swaps or futures so the portfolio's exposure to each factor is offset.

The chapter links to bond valuation, spot and forward rates, and futures and swaps in Financial Markets and Products. It also connects to the quantitative analysis material on variance, covariance and factor models. The same skills appear in market risk measurement, where curve risk feeds VaR.

Questions here are numeric and reward a clear method. A single parallel-shift hedge can fail, and the exam likes to test exactly why and how a multi-factor hedge fixes it. If you can set up the exposure equations, solve for hedge amounts and read PCA output, you can pick up reliable marks on a topic many candidates only skim. The ideas also support later study of interest rate risk, so the effort pays off beyond this chapter.

Modeling Non-Parallel Term Structure Shifts and Hedging: topics in the order to study them

1Duration, DV01 and Convexity ReviewEvery later idea is a refinement of these parallel-shift measures, so lock in the formulas and units first.
2Key Rate Exposures and Key Rate ShiftsThis is the first step beyond parallel shifts and gives you the exposure-by-maturity vector used in hedging.
3Principal Components Analysis of Term StructurePCA replaces many key rates with a few factors, and you need key rate thinking to see why that helps.
4Multi-Factor Hedging with Swaps and FuturesHere you combine exposures from the earlier topics and solve for hedge positions across several instruments.
5Bucketing, Hedge Effectiveness and Curve TradesIt closes the chapter by judging how well a hedge works and applying the tools to trades on curve shape.

How to prepare Modeling Non-Parallel Term Structure Shifts and Hedging

Treat this chapter as a build-up. Each topic adds one layer to the same idea, so work in order and practise calculations by hand.

Rewrite the duration, DV01 and convexity formulas from memory, with the price-change approximation using both duration and convexity. Check the units and sign each time.
Work a key rate example on paper. Shift one maturity point by 1 basis point, record the price change, repeat for each point, and check that the key rate exposures sum to roughly the parallel DV01.
Learn what PCA output looks like: factor loadings by maturity, variance explained, and what level, slope and curvature mean. Practise stating the effect of a one-unit factor move on a portfolio.
Set up hedge problems as simultaneous equations. Write one equation per factor or key rate, put the unknown hedge quantities on one side, and solve. Use a financial calculator or the matrix-free substitution method.
Practise a mix of single-factor and multi-factor hedges, then compare the residual exposure each leaves under a twist or steepening.
Finish with timed sets of mixed questions, aiming for about two minutes each, and keep an error log for sign and unit slips.

Common mistakes in Modeling Non-Parallel Term Structure Shifts and Hedging

Hedging only with a single DV01 match and calling the portfolio hedged against all curve moves.
Fix: Check exposure at each key rate or factor. Ask what happens under a steepening or twist before accepting a hedge.
Mixing signs when solving hedge equations, so the hedge adds exposure instead of removing it.
Fix: Define signs before you start. Write the portfolio exposure, the instrument exposures and set the total to zero, then check the answer by substitution.
Using basis point shifts as decimals without conversion, such as treating 1 bp as 0.01.
Fix: Remember 1 bp = 0.0001 = 0.01%. Convert the shift first, then apply it.
Assuming PCA factors are fixed economic variables with the same meaning for every portfolio.
Fix: Treat them as statistical factors estimated from data. Their labels describe the typical loading pattern across maturities.
Using duration alone for large yield moves and ignoring convexity.
Fix: Add the convexity term when the question gives a large shift or asks for the more accurate price change.
Forgetting that key rate exposures are measured with other rates held fixed, then double-counting when summing them.
Fix: Read each exposure as a one-point-at-a-time effect. Sum them only to compare with the parallel-shift measure.

Last-day revision: Modeling Non-Parallel Term Structure Shifts and Hedging

DV01 is the price change for a 1 basis point fall in yield, quoted as a positive number for a long bond position.
DV01 ≈ modified duration × price × 0.0001.
Price change ≈ −D × P × Δy + ½ × C × P × (Δy)², with D as modified duration.
Convexity helps a long bond: it adds to gains when yields fall and cushions losses when yields rise.
Key rate exposures measure sensitivity to a shift at one maturity with other key rates held fixed.
Key rate exposures sum to approximately the parallel-shift exposure.
PCA factors are uncorrelated, and the first few typically explain most of the curve variation.
The usual interpretations are level (first), slope (second) and curvature (third).
A multi-factor hedge needs at least as many hedge instruments as factors you want to neutralise.
Write one exposure equation per factor and solve them together for the hedge amounts.
A hedge that is neutral to a parallel shift can still lose money under a steepening or twist.
Hedge effectiveness is judged by the residual exposure left after the hedge.

Modeling Non-Parallel Term Structure Shifts and Hedging practice questions

Modeling Non-Parallel Term Structure Shifts and Hedging in other exams

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

Modeling Non-Parallel Term Structure Shifts and Hedging: frequently asked questions

What is a non-parallel shift of the yield curve?

It is a change where rates at different maturities move by different amounts. Common examples are steepening, flattening and twists. Duration alone cannot capture these moves.

How are key rate exposures different from DV01?

DV01 measures the effect of the whole curve moving by 1 basis point. A key rate exposure measures the effect of moving one maturity point while others stay fixed. Together the key rate exposures roughly add up to the DV01.

Do I need to run PCA myself in the FRM exam?

No. You are expected to interpret PCA results, such as loadings and variance explained, and understand how they are used in hedging. Focus on meaning and application rather than computation.

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

You need at least as many instruments as the factors or key rates you want to neutralise. With two exposures to offset, you use two hedging instruments and solve two equations together.

Can I solve these hedge questions with a financial calculator?

A calculator helps with bond prices, yields and DV01 inputs. The hedge itself is usually two or three linear equations, which you can solve by substitution on paper.