FRM Exam Part II · Regression Hedging and Principal Component Analysis

Regression Hedging and Hedge Ratios Explained

Q: What is the difference between a regression hedge and a DV01 hedge?

A DV01 hedge matches the dollar sensitivity to a one bp move and assumes both yields move one-for-one. A regression hedge multiplies that hedge by the estimated β, so it allows for yields that move by different amounts.

Q: Why regress yield changes and not yield levels?

Yield levels trend and are strongly autocorrelated, which can give a high R² and an unreliable slope. Changes are closer to stationary, and hedging is about changes in value, so changes are the right input.

Q: What does R-squared tell me about hedge effectiveness?

It is the share of the variance of the position's yield changes explained by the hedge's yield changes. The unexplained 1 − R² is basis risk that remains after hedging.

Q: Is a regression hedge better than a DV01 hedge?

It minimises variance based on past data, so it usually fits history better. But β can change over time, so the improvement is not guaranteed out of sample.

Updated 11 October 2026 · Fact-checked

A regression hedge regresses changes in the yield of the position you hold on changes in the yield of the hedging instrument. The slope β adjusts the DV01 hedge: Hedge face = (DV01 of position ÷ DV01 of hedge) × β × position face. R-squared measures how much yield variance the hedge removes.

Understand Regression Hedging and Hedge Ratios

A DV01 hedge assumes the yield on your position and the yield on your hedging instrument move one-for-one. In practice they do not. A corporate bond yield may move 1.2 bp for every 1 bp move in a Treasury yield. A one-for-one assumption then under-hedges.

A regression hedge fixes this using history. You collect daily or weekly changes in the yield of the position (the dependent variable, Δy_position) and changes in the yield of the hedge instrument (the independent variable, Δy_hedge). You fit Δy_position = α + β × Δy_hedge + ε. The slope β is the yield beta. It says how many bp the position yield tends to move per 1 bp move in the hedge yield.

The hedge ratio then scales the DV01 hedge by β. If β is 1.2, you need 20% more hedge DV01 than a plain DV01 hedge. The idea: the position's P&L per bp of hedge-yield move is DV01_position × β, so the hedge must offset that amount.

The regression also gives R-squared, the share of variance in the position's yield changes explained by the hedge's yield changes. The remaining part, 1 − R², is the residual (basis) risk that the hedge cannot remove. The regression minimises the variance of the hedged position's P&L, so it is called a variance-minimising hedge. Past relationships can break down, so the result is an estimate, not a guarantee.

This is the two-variable regression hedge, as in Tuckman. Hedges with several instruments need multiple regression or factor methods such as PCA.

Key formulas to remember

Regression model: Δy_position = α + β × Δy_hedge + ε; Use changes in yield, not yield levels. Position is the dependent variable.
Yield beta: β = Cov(Δy_position, Δy_hedge) ÷ Var(Δy_hedge) = ρ × σ_position ÷ σ_hedge; Slope of the regression line. Units are bp per bp.
Regression hedge face amount: F_hedge = F_position × (DV01_position ÷ DV01_hedge) × β; DV01 values per unit of face (e.g. per ₹/$ 100 face). Sign: take the opposite side to the position.
Hedge DV01 required: DV01_hedge total = β × DV01_position total; Equivalent way to size the hedge when you know total DV01.
Hedge effectiveness: R² = 1 − Var(residual) ÷ Var(Δy_position) = ρ²; In a two-variable regression, R² is the square of the correlation.
Residual standard deviation of yield change: σ_ε = σ_position × √(1 − R²); Unhedged basis risk in yield-change terms. Multiply by DV01 for P&L risk.

How to solve Regression Hedging and Hedge Ratios questions

Use this method for any question on regression hedges, whether it gives you β directly, or gives correlation and volatilities.

1Identify the position (dependent variable) and the hedging instrument (independent variable). Confirm the data are yield changes.
2Get β. Use the stated slope, or compute ρ × σ_position ÷ σ_hedge, or Cov ÷ Var(hedge).
3Compute or read the DV01 of the position and of the hedge instrument. Keep units consistent (per 100 face or total).
4Compute the hedge DV01 needed: β × DV01_position. Then divide by the hedge DV01 per unit to get the face amount or number of contracts.
5Set the sign. The hedge is the opposite side to the position: short the hedge if you are long the position.
6If asked about effectiveness, use R² = ρ² and residual risk = σ_position × √(1 − R²), converted to money with DV01.
7Compare with the plain DV01 hedge if asked: the regression hedge equals the DV01 hedge times β.
8Interpret in one line: what risk is removed, and what basis risk remains.

Quickest way: Scale the DV01 hedge by β

When to use it: Any numerical question where you are given β (or ρ and the two volatilities) and DV01s. Do this first, then check the options.

Compute the plain DV01 hedge amount first.
Multiply it by β. If β > 1, the hedge gets bigger; if β < 1, smaller.
If β is not given, β = ρ × σ_position ÷ σ_hedge.
For effectiveness, square ρ to get R². Residual volatility = σ_position × √(1 − R²).
Eliminate options with the wrong direction or that ignore β.

Common mistakes in Regression Hedging and Hedge Ratios

Regressing yield levels instead of yield changes.
Level data look smooth and give a very high R², so it feels right.
Fix: Always use changes in yield. Levels are trending and give misleading slopes and R².
Swapping the dependent and independent variables, so β = ρ × σ_hedge ÷ σ_position.
The formula looks symmetric and students pick the wrong ratio.
Fix: Position yield change is the dependent variable. β = ρ × σ_position ÷ σ_hedge.
Multiplying the DV01 ratio by β but forgetting the DV01 ratio itself (or the reverse).
Students remember 'adjust by β' and stop there.
Fix: Write F_hedge = F_position × (DV01_position ÷ DV01_hedge) × β in full each time.
Reading R² as the share of P&L hedged in money terms or as 'the hedge works R² of the time'.
R² is described loosely as effectiveness.
Fix: R² is the share of yield-change variance explained. Residual volatility is σ × √(1 − R²), so risk falls by less than R² suggests.
Taking the same side as the position.
Focus on the arithmetic and ignore direction.
Fix: A hedge offsets. Long position means short hedge.
Assuming the estimated β will hold in future.
The regression gives a precise-looking number.
Fix: Treat β as an estimate from a sample. It can shift in stress and with the choice of data window.

Worked examples

Example 1

You hold a long position in corporate bonds of face $50 million with a DV01 of $0.0650 per $100 face. You hedge with Treasury futures-equivalent bonds with DV01 of $0.0800 per $100 face. Regressing weekly changes in the corporate yield on changes in the Treasury yield gives β = 1.15. What face amount of the Treasury hedge do you take and on which side?

Show the solution

Position DV01 per $100 face = 0.0650; hedge DV01 per $100 face = 0.0800.
DV01 ratio = 0.0650 ÷ 0.0800 = 0.8125.
Plain DV01 hedge face = $50 million × 0.8125 = $40.625 million.
Regression hedge face = $40.625 million × 1.15 = $46.71875 million.
You are long the corporate bonds, so you short the Treasuries.

Answer: Short about $46.72 million face of the Treasury hedge, which is 15% more than the plain DV01 hedge of $40.625 million.

Example 2

Weekly yield changes of a bond portfolio have a standard deviation of 8 bp. The hedge instrument's yield changes have a standard deviation of 10 bp. The correlation is 0.90. The portfolio DV01 is ₹4,00,000 per bp. Find β, the hedge DV01 needed, R², and the residual weekly standard deviation of the hedged position in rupees (to the nearest thousand).

Show the solution

β = ρ × σ_position ÷ σ_hedge = 0.90 × 8 ÷ 10 = 0.72.
Hedge DV01 needed = 0.72 × ₹4,00,000 = ₹2,88,000 per bp of hedge-yield move.
R² = 0.90² = 0.81.
Residual yield-change SD = 8 × √(1 − 0.81) = 8 × √0.19 = 8 × 0.43589 = 3.487 bp.
Residual P&L SD = 3.487 × ₹4,00,000 = ₹13,94,800, about ₹13,95,000.

Answer: β = 0.72; hedge DV01 ₹2,88,000 per bp; R² = 0.81; residual weekly SD of about ₹13,95,000 (versus ₹32,00,000 unhedged).

Exam tips

Check which yield is the dependent variable. Questions often hide this in the wording.
Expect to scale a DV01 hedge by β. Compute the DV01 hedge first, then multiply.
When given ρ and volatilities, you need β = ρ × σ_position ÷ σ_hedge. Do not use R² as β.
For effectiveness questions, residual risk uses √(1 − R²), not 1 − R².
In interpretation questions, mention basis risk and that β is estimated from historical data and may be unstable.

Practice questions from Regression Hedging and Principal Component Analysis

Regression Hedging and Hedge Ratios 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 Hedge Ratios: frequently asked questions

What is the difference between a regression hedge and a DV01 hedge?

A DV01 hedge matches the dollar sensitivity to a one bp move and assumes both yields move one-for-one. A regression hedge multiplies that hedge by the estimated β, so it allows for yields that move by different amounts.

Why regress yield changes and not yield levels?

Yield levels trend and are strongly autocorrelated, which can give a high R² and an unreliable slope. Changes are closer to stationary, and hedging is about changes in value, so changes are the right input.

What does R-squared tell me about hedge effectiveness?

It is the share of the variance of the position's yield changes explained by the hedge's yield changes. The unexplained 1 − R² is basis risk that remains after hedging.

Is a regression hedge better than a DV01 hedge?

It minimises variance based on past data, so it usually fits history better. But β can change over time, so the improvement is not guaranteed out of sample.