FRM Exam Part II · Regression Hedging and Principal Component Analysis

PCA-Based Hedging and Risk Measurement Explained

Updated 11 October 2026 · Fact-checked

PCA hedging replaces many rate moves with a few uncorrelated factors, usually level, slope and curvature. You compute your portfolio's exposure to each factor, then add hedge instruments so those exposures sum to zero. For risk, you square each exposure, multiply by factor variance, add them up, take the root, and scale by a z-value for VaR.

Understand PCA-Based Hedging and Risk Measurement

A yield curve has many points, but they do not move independently. Principal component analysis (PCA) takes the historical changes in many rates and finds a small set of uncorrelated factors that explain most of the movement. For government curves, the first three are usually level (all rates move together), slope (short and long rates move in opposite directions) and curvature (the middle moves against the wings).

Each factor has two key numbers. The loading tells you how many basis points each rate moves for one unit of the factor. The variance (the eigenvalue) tells you how much that factor moves in a typical period. The share of total variance a factor explains is its eigenvalue divided by the sum of all eigenvalues.

To measure your factor exposure, take the position's sensitivity at each rate (its key-rate DV01) and multiply by that rate's loading on the factor. Add across all rates. The result is the money you gain or lose for a one-unit move in that factor. Do this for each factor you keep.

A PCA hedge chooses hedge positions so that the total exposure to each retained factor is zero. You need one hedging instrument for each factor you want to neutralize. Neutralizing level and slope needs two instruments, and adding curvature needs three. This is a system of simple linear equations.

A regression hedge is different. It regresses the yield change on one rate against the yield change on another and sizes the hedge from the fitted slope. It targets one relationship and depends on the sample. A PCA hedge targets the common factors across the whole curve, so it is built around systematic curve shapes. Neither hedge removes risk from factors you leave out.

PCA also gives a clean risk measure. Because the factors are uncorrelated, portfolio variance is just the sum of (exposure² × factor variance). You need no correlation matrix. Take the square root for volatility, and multiply by a z-value for a parametric VaR. The risk from dropped components is ignored, so the figure is slightly understated.

Key formulas to remember

Factor exposure to PC k: βk = Σj (DV01j × loading of PC k on rate j); Use one sign convention for DV01 and loadings throughout. The sum runs over every rate where the portfolio has a key-rate sensitivity.
PCA hedge condition: βk(portfolio) + βk(hedge) = 0 for each retained PC k; You need as many hedge instruments as factors you neutralize. Solve the equations together.
Variance explained by a PC: % explained by PC k = λk ÷ Σ λ; λk is the eigenvalue (variance) of component k. Component variances sum to total variance.
Portfolio variance using PCs: σp² = Σk (βk² × σk²); Valid because PCs are uncorrelated. σk is the volatility of factor k in the same units as the loadings. Dropped PCs are ignored.
Portfolio volatility: σp = √(Σk βk² σk²); Take the root only after summing. Do not add exposures times volatilities directly.
Parametric PCA VaR: VaR = z × σp; For a normal model, z is 1.645 at 95% and 2.326 at 99% one-tailed. Scale by √t for a t-day horizon if the factor volatility is daily.

How to solve PCA-Based Hedging and Risk Measurement questions

Use this order for any PCA hedging or PCA risk question. It keeps units and signs under control.

1Write down the factors given (level, slope, curvature) and the loading of each factor on each rate.
2List the portfolio's key-rate DV01s, using one sign convention. Note which rates the hedge instruments sit at.
3Compute the portfolio's exposure to each factor: multiply each DV01 by its loading and add across rates.
4For a hedge, let the unknown hedge DV01s be x1, x2 and so on. Write one equation per factor: portfolio exposure plus hedge exposure equals zero.
5Solve the equations. Check by substituting back into every factor equation.
6For risk, square each remaining exposure, multiply by that factor's variance, and add. Take the square root for volatility.
7Multiply by the z-value (and by √t if the horizon is longer) to get VaR.
8State what is left unhedged or ignored: higher-order factors, and non-parallel risk outside the retained PCs.

Quickest way: Exposure table and two equations

When to use it: Use when the question gives loadings for two or three rates and asks for hedge sizes or factor-based volatility.

Build a small grid with rates as rows and factors as columns. Fill in the loadings.
Compute portfolio exposure per factor first. If a factor loading at the portfolio's rate is zero, that exposure is zero.
Write the hedge equations. If a loading is zero for one hedge instrument, solve that equation first to cut the work.
Add the two equations to eliminate one unknown, as the loadings are often symmetric.
For VaR, compute each term β × σ first, square them, add, then root. Check that a large level term dominates; if so, your answer should be close to it.

Common mistakes in PCA-Based Hedging and Risk Measurement

Adding exposures times volatilities instead of squaring and summing.
Students treat the factors as perfectly correlated, as in simple DV01 math.
Fix: PCs are uncorrelated. Use σp = √(Σ βk² σk²). A quick sense check: the result must be less than the plain sum of the terms.
Hedging with fewer instruments than factors.
It is tempting to use one bond to cover level and slope together.
Fix: One instrument neutralizes one factor. Two factors need two instruments with different loadings. Otherwise the equations cannot all be satisfied.
Mixing signs between DV01 and loadings.
Slope loadings are negative at the short end, and DV01 can be defined as loss or gain per rise or fall.
Fix: Fix one convention before starting. Carry negative loadings through the arithmetic and check each equation sums to zero after solving.
Treating a PCA hedge and a regression hedge as the same thing.
Both give a hedge ratio from historical data.
Fix: A regression hedge fits one rate's change against another's and minimizes variance for that pair. A PCA hedge zeros exposure to common curve factors across all rates. Say which one the question uses.
Assuming a PCA hedge removes all interest rate risk.
Zero exposure to the chosen factors feels like zero risk.
Fix: Exposure to omitted components and to rate moves outside the PCA sample remains. The hedge is only as good as the factor set and the stability of the loadings.
Using eigenvalue as volatility.
The eigenvalue and the factor's standard deviation are easy to mix up.
Fix: The eigenvalue is the variance. Take its square root for the factor's volatility before multiplying by exposure, or use exposure² × eigenvalue directly.

Worked examples

Example 1

A USD portfolio gains USD 100,000 per 1 bp fall in the 7-year rate and USD 20,000 per 1 bp fall in the 10-year rate. Factor loadings (bp move per one unit of the factor) are: Level: 5y 1.0, 7y 1.0, 10y 1.0. Slope: 5y −0.5, 7y 0, 10y +0.5. Hedge with 5-year and 10-year instruments so that exposure to both level and slope is zero. Find the hedge DV01s, using the same sign convention (positive = gains when rates fall).

Show the solution

Level exposure of the portfolio = 100,000 × 1.0 + 20,000 × 1.0 = 120,000.
Slope exposure of the portfolio = 100,000 × 0 + 20,000 × 0.5 = 10,000.
Let x5 and x10 be the hedge DV01s at 5y and 10y.
Level equation: 120,000 + x5 + x10 = 0.
Slope equation: 10,000 + (−0.5)x5 + 0.5x10 = 0, so x10 − x5 = −20,000, giving x10 = x5 − 20,000.
Substitute: 120,000 + x5 + x5 − 20,000 = 0, so 2x5 = −100,000 and x5 = −50,000. Then x10 = −70,000.
Check level: 120,000 − 50,000 − 70,000 = 0. Check slope: 10,000 + 25,000 − 35,000 = 0.

Answer: Take a short 5-year position with DV01 of USD 50,000 and a short 10-year position with DV01 of USD 70,000. Both level and slope exposures are then zero.

Example 2

Using the unhedged portfolio above, the level exposure is USD 120,000 per unit and the slope exposure is USD 10,000 per unit. The daily factor volatilities are 6 units for level and 3 units for slope. Ignore other components and assume normality. Estimate the one-day 99% PCA-based VaR (z = 2.326).

Show the solution

Level term: 120,000 × 6 = 720,000.
Slope term: 10,000 × 3 = 30,000.
Portfolio variance = 720,000² + 30,000² = 518,400,000,000 + 900,000,000 = 519,300,000,000.
Volatility = √519,300,000,000 ≈ 720,625. Check: 720,000² is 518.4 billion, and the extra 0.9 billion adds about 625.
VaR = 2.326 × 720,625 ≈ 1,676,000.
Note that the slope term adds very little because the level exposure dominates.

Answer: One-day 99% VaR ≈ USD 1.68 million, based on level and slope only. Risk from curvature and other components is excluded, so the true figure is slightly higher. After the hedge above, the VaR from these two factors falls to zero.

Exam tips

Questions usually give loadings and ask for either hedge sizes or portfolio volatility. Identify which one first, then use the matching formula.
Check the number of factors against the number of hedge instruments. If they do not match, the question is probably testing whether a full neutralization is possible.
When asked to compare a PCA hedge with a regression hedge, answer on what each targets: factor exposures across the curve versus the fit between two rates.
Expect conceptual options about what PCA misses: omitted components, unstable loadings, and moves not seen in the sample. A hedge with zero exposure on retained factors is not riskless.
On the VaR calculation, do the squaring and summing before the square root. Use the dominant term as a sense check on your answer.

Practice questions from Regression Hedging and Principal Component Analysis

PCA-Based Hedging and Risk Measurement: frequently asked questions

How do I hedge level and slope exposure using PCA?

Compute your exposure to the level and slope factors from your key-rate DV01s and the loadings. Then choose two hedge instruments at different maturities and solve two equations so total level exposure and total slope exposure are both zero. The instruments must have different slope loadings, or the equations have no solution.

What is the difference between a PCA hedge and a regression hedge?

A regression hedge sizes a position from the fitted relationship between yield changes at two points, so it targets one pair of rates. A PCA hedge sets exposure to common factors such as level and slope to zero across the whole curve. PCA hedges are built around systematic curve shapes, while regression hedges depend on the chosen pair and sample.

How is factor exposure to a principal component calculated?

Multiply the portfolio's DV01 at each rate by that rate's loading on the component, then add across all rates. The sum is the profit or loss for a one-unit move in that component. Use one consistent sign convention for DV01 and loadings.

How is PCA used to estimate VaR for an interest rate portfolio?

Compute the portfolio's exposure to each retained factor. Because the factors are uncorrelated, portfolio variance equals the sum of exposure squared times factor variance. Take the square root for volatility, then multiply by the z-value for the confidence level. The result ignores any components left out.