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FRM Part II · FRM Exam Part II

Regression Hedging and Principal Component Analysis for FRM Part II

Regression hedging and PCA are ways to hedge interest rate risk when yields do not move in parallel. DV01 hedging matches price sensitivity. Regression uses historical yield changes to set the hedge ratio. PCA reduces many yield moves to a few factors: level, slope and curvature, then hedges each one.

What this chapter covers

This chapter is about hedging fixed income positions when the yield curve does not shift in parallel. It starts with DV01, the price change for a one basis point move in yield. It then moves to regression hedging, where you estimate how much the yield on your position moves for each unit of move in the yield on the hedging instrument. It ends with principal component analysis (PCA), which describes curve moves with a small set of independent factors.

The logic is a progression. DV01 hedging assumes one common yield change. Regression hedging relaxes this by allowing the two yields to move by different amounts, but it still uses one risk factor. PCA allows several factors, usually level, slope and curvature, so you can hedge or measure risk against each one.

This chapter links to the rest of the paper in several places. It supports market risk measurement, where you must understand how a portfolio responds to yield curve moves. It connects to the term structure and key rate material, to VaR on fixed income books, and to the treasury and liquidity topics, where banks hedge interest rate risk in the banking book. Questions are usually applied, so expect to compute a hedge, then interpret it.

Hedging and factor questions are quantitative but short, which makes them reliable marks if you know the method. They reward candidates who can set up the hedge ratio, get the sign right and explain what risk remains. The ideas also carry into other Part II topics, so the effort pays back more than once. With 80 equally weighted questions in 4 hours, a chapter where you can answer fast and accurately saves time for harder case questions.

Regression Hedging and Principal Component Analysis: topics in the order to study them

  1. 1DV01 and Duration-Based HedgingIt is the base case. Every later method is a refinement of matching price sensitivity to yield moves.
  2. 2Regression Hedging and Hedge RatiosIt fixes the main weakness of DV01 hedging, the assumption that both yields move one-for-one, using a single estimated beta.
  3. 3Level, Slope and Curvature FactorsYou need the intuition for how curves actually move before you meet the statistical tool that extracts those moves.
  4. 4Principal Component Analysis FundamentalsIt shows how the factors are built from the covariance or correlation of yield changes and how much variance each explains.
  5. 5PCA-Based Hedging and Risk MeasurementIt combines everything. You hedge exposure to each factor and use factor volatility to measure portfolio risk.

How to prepare Regression Hedging and Principal Component Analysis

Build the chapter in layers. Make each method automatic before you add the next, and always finish by asking what risk is left over.

  1. Practise DV01 calculations until you can compute the hedge amount in one line: face value of hedge = DV01 of position ÷ DV01 of hedge per unit, with the correct sign.
  2. Learn the regression hedge logic. The hedge ratio is the regression slope of the position's yield change on the hedge instrument's yield change, and you apply it to the DV01 hedge. Check what is the dependent variable before you compute.
  3. Draw the three standard factor shapes and say in words what a one unit move does to short, medium and long yields.
  4. Study PCA as a variance story. Components are uncorrelated, ordered by variance explained, and the first few usually explain most of the curve variation.
  5. Work hedging questions with two or three instruments, solving for each factor exposure so the net exposure to each factor is zero.
  6. Finish with mixed practice questions under time. For each, write the risk measure, the method and the interpretation, as the exam expects.

Common mistakes in Regression Hedging and Principal Component Analysis

  • Getting the direction of the hedge wrong

    Fix: Write the sign of the position exposure first, then take the opposite exposure in the hedge.

  • Reversing the regression variables

    Fix: Put the yield change of the position you are hedging on the left side, and the hedging instrument on the right.

  • Treating DV01 hedging as protection against any curve move

    Fix: Say that it protects against that one assumed move, and that slope and curvature risk can remain.

  • Confusing factors with maturities

    Fix: Treat each factor as a pattern of moves across all maturities, and describe its effect on short, medium and long yields.

  • Assuming PCA factors have fixed economic meaning

    Fix: Remember that PCA finds statistical components from data, and the labels come from how the loadings look.

  • Ignoring residual risk after the hedge

    Fix: Always ask which factors were left out and whether the historical relationship might change.

Last-day revision: Regression Hedging and Principal Component Analysis

  • DV01 is the price change for a one basis point change in yield.
  • A DV01 hedge makes the position's and hedge's price changes offset for the same yield change.
  • Hedge face amount = position DV01 ÷ hedge DV01 per unit of face value, and the hedge goes the opposite way.
  • Regression hedge ratio is the slope of position yield change on hedge yield change, so it scales the DV01 hedge.
  • A regression hedge still relies on one factor and on past relationships holding.
  • Level means a parallel shift, slope means steepening or flattening, curvature means belly versus wings.
  • PCA turns correlated yield changes into uncorrelated components.
  • Components are ranked by variance explained, and level is usually the largest.
  • PCA hedging sets net exposure to each chosen factor to zero.
  • Factor-based risk uses each factor's exposure and its volatility, and uncorrelated factors let variances add.
  • Residual risk is what remains after the chosen factors are hedged.

Regression Hedging and Principal Component Analysis practice questions

Regression Hedging and Principal Component Analysis in other exams

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

Regression Hedging and Principal Component Analysis: frequently asked questions

What is the difference between DV01 hedging and regression hedging?

DV01 hedging matches price sensitivity and assumes both yields change by the same amount. Regression hedging adjusts the hedge using an estimated relationship between the two yield changes. It is more realistic but depends on historical data.

Why does PCA matter for hedging?

Yield curve moves across maturities are highly correlated. PCA reduces them to a few uncorrelated factors, so you can hedge exposure to each factor instead of to every maturity separately.

Do I need to compute PCA by hand in the exam?

Questions usually focus on interpreting components, variance explained and factor exposures rather than full eigenvector calculations. You should still be able to use given loadings to compute exposures and hedges.

How should I study this chapter on my phone?

Keep a short sheet of the hedge formulas and factor shapes for quick review. Do the calculation practice on paper, because setting up the signs and variables is where marks are lost.