Regression Hedging and Principal Component Analysis: formula sheet

Key formulas

DV01 from price

DV01 = −(P(y + 0.0001) − P(y − 0.0001)) ÷ 2

Central difference is the usual numerical definition. A simple version uses the price change for a 1 bp move.

DV01 from modified duration

DV01 ≈ ModD × P × 0.0001

P is the full (dirty) price of the position in currency. ModD is modified duration in years.

Modified duration

ModD = Macaulay duration ÷ (1 + y ÷ m)

m is compounding periods per year. Use y ÷ m for the periodic yield.

DV01 hedge ratio (face value)

Hedge face value = (DV01 of position ÷ DV01 of hedge per 100 face) × 100

The hedge is taken in the opposite direction to the position.

Hedge units

N = DV01 position ÷ DV01 per unit of hedge instrument

For futures, use DV01 per contract (often the cheapest-to-deliver DV01 divided by its conversion factor).

Duration-based hedge

Hedge value = Position value × ModD position ÷ ModD hedge

Equivalent to DV01 matching when the position and hedge are priced and compared on the same basis.

Price change estimate

ΔP ≈ −DV01 × Δy (in bp)

Valid for small parallel moves. Ignores convexity.

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.

Rate change from factors

Δr(T) ≈ L₁(T)·F₁ + L₂(T)·F₂ + L₃(T)·F₃

L are loadings at maturity T and F are factor moves (factor scores). Remaining components are ignored.

Variance explained by a component

Share of variance = λᵢ ÷ Σλⱼ

λ are eigenvalues of the covariance matrix. Cumulative share is the sum of the first k shares.

Sign pattern for naming factors

Level: same sign everywhere. Slope: one sign change (short vs long). Curvature: two sign changes (ends vs middle).

Signs of a whole component can be flipped without changing its meaning. Read the shape, not the absolute sign.

Portfolio exposure to a factor

Factor exposure = Σ DV01(T) × L(T)

Sum over key rate maturities. Gives the P&L sensitivity to a one-unit move in that factor.

Eigenvalue equation

Σv = λv

Σ is the covariance matrix, v is an eigenvector (loadings), λ is its eigenvalue, the variance explained by that component.

Variance explained by component k

λk ÷ (λ1 + λ2 + ... + λn)

Multiply by 100 for a percentage. The denominator equals the total variance, the sum of the diagonal of Σ.

Cumulative variance explained

(λ1 + ... + λk) ÷ Σλ

Used to decide how many components to keep.

Total variance

Trace(Σ) = Σ of the variances = Σ of the eigenvalues

Use it to check that your eigenvalues are consistent.

Component score

PCk = v1k·x1 + v2k·x2 + ... + vnk·xn

The xi are the changes in the original factors. Scores of different components are uncorrelated.

Orthogonality and normalisation

vi · vj = 0 for i ≠ j; Σ(vik)² = 1

Loadings are scaled so each eigenvector has unit length.

Volatility of a component

σk = √λk

Standard deviation of the component score.

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.

Quick revision

• 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.

Common mistakes

• Using Macaulay duration instead of modified duration in the DV01 formula. Fix: Convert first: ModD = Macaulay ÷ (1 + y ÷ m). Then compute DV01.
• Hedging with equal face values rather than equal DV01. Fix: A 10-year bond has a larger DV01 than a 2-year bond of the same face value. Always match DV01.
• Regressing yield levels instead of yield changes. 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. Fix: Position yield change is the dependent variable. β = ρ × σ_position ÷ σ_hedge.
• Calling the first component a perfectly parallel shift. Fix: The level loadings are same-signed and similar, not always identical. Say roughly parallel unless the question says exact.
• Treating a negative loading as a falling rate. Fix: Rate change = loading × factor move. A negative loading with a negative factor move gives a rise in that rate.
• Dividing an eigenvalue by the largest eigenvalue, or by the number of variables, instead of by the sum of eigenvalues. Fix: Always use eigenvalue ÷ sum of all eigenvalues. The sum equals the trace of the covariance matrix.
• Saying the eigenvector gives the variance explained. Fix: The eigenvector holds the weights (direction). The eigenvalue is the variance along that direction.
• Adding exposures times volatilities instead of squaring and summing. 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. Fix: One instrument neutralizes one factor. Two factors need two instruments with different loadings. Otherwise the equations cannot all be satisfied.

Exam tips

• Read whether the question gives price per 100 or total market value. Many wrong answers come from unit mismatch.
• Expect a conceptual MCQ on limits: a DV01 hedge fails for non-parallel shifts, large moves and convexity differences.
• Check the direction word (short, long) before you pick an option.
• When a question asks for the difference between DV01 and modified duration, answer: DV01 is currency per bp, duration is percent per 100 bp.
• For hedged P&L after a non-parallel move, expect residual loss or gain; do not assume zero.
• 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.