FRM Part I · FRM Exam Part I

Modeling Non-Parallel Term Structure Shifts and Hedging: formula sheet

Full chapter guide

Key formulas

Modified duration from Macaulay duration: D_mod = D_Mac ÷ (1 + y/m); y is the annual yield, m is compounding periods per year. With continuous compounding, modified duration equals Macaulay duration.
First-order price change: ΔP ÷ P ≈ −D_mod × Δy; Δy in decimals. A 25 bp move is 0.0025.
DV01: DV01 = D_mod × P × 0.0001; Quoted as a positive number per 1 bp. The price falls by this amount when yield rises 1 bp. Here P is the full (dirty) price of the position.
Duration and convexity approximation: ΔP ÷ P ≈ −D_mod × Δy + ½ × C × (Δy)²; C is convexity. Convexity is positive for plain bonds, so the second term adds to price.
Portfolio duration: D_p = Σ (w_i × D_i); w_i are market value weights. Portfolio DV01 is simply the sum of position DV01s.
DV01 hedge ratio: N = −DV01_target ÷ DV01_hedge; N is the number of hedge instrument units. The negative sign means take the opposite position.
Key rate 01: KR01(k) = −[V(+1 bp at key rate k) − V(−1 bp at key rate k)] ÷ 2, or −[V(+1 bp) − V₀] for a one-sided shift; Only key rate k moves, with linear interpolation to neighbouring key rates. A positive KR01 means value falls when that rate rises.
Key rate duration: KRD(k) = −[V(+Δ) − V(−Δ)] ÷ (2 × V₀ × Δ); Δ is the shift in decimal form, so 1 bp = 0.0001. Equivalent to KR01 ÷ (V₀ × 0.0001).
Relation of KR01 to KRD: KR01(k) = KRD(k) × V₀ × 0.0001; Use this to switch between the two measures.
Sum rule: Σ KR01(k) = DV01 and Σ KRD(k) = effective duration; Holds when the key rate shifts together make a parallel shift of 1 bp. For a portfolio, add KR01s across positions at each key rate.
Approximate price change: ΔV ≈ −Σ KR01(k) × Δy(k), with Δy in bp; Use this to estimate the value change for a non-parallel move.
Variance explained by a factor: Share of factor i = λᵢ ÷ Σλⱼ; λᵢ is the eigenvalue of factor i. The denominator is the total variance, the sum of all eigenvalues.
Cumulative variance explained: Cumulative share (first k factors) = (λ₁ + … + λₖ) ÷ Σλⱼ; Add the shares of the top k factors. They are uncorrelated, so variances add.
Total variance: Σλⱼ = Σ variance of each rate change (when PCA uses the covariance matrix); With a correlation matrix, the total equals the number of variables.
Factor standard deviation: σᵢ = √λᵢ; The eigenvalue is a variance. Take the square root for a volatility in the same units as the rate changes.
Rate change from factors: Δrate(T) = Σ (loading of factor i at T × factor score i); Loadings come from eigenvectors. The factor scores are uncorrelated.
Eigenvector normalization: Σ(loadings of one factor)² = 1; Use this to check or fill in a missing loading.
Key rate 01 of a position: KR01(k) = change in value for a 1 bp fall in key rate k, with other key rates unchanged; Sign convention matters. State whether exposure is per rate fall or per rate rise and keep it the same for the portfolio and the hedges.
Sum of key rate exposures: Σ KR01(k) ≈ DV01; Key rate exposures add up to the parallel-shift DV01. Use this as a quick check on your inputs.
Multi-factor hedge condition: E_P(j) + Σ N_i × E_i(j) = 0 for every factor j; E_P(j) is portfolio exposure to factor j, E_i(j) is exposure of one unit of instrument i, N_i is the position in instrument i. Solve for all N_i together.
Two-factor, two-instrument solution: N_B = −E_P(2) ÷ E_B(2) if instrument A has no exposure to factor 2; then N_A = −(E_P(1) + N_B × E_B(1)) ÷ E_A(1); Works when instruments are triangular, with A touching only factor 1. Otherwise solve the two equations by substitution.
Factor exposure of a portfolio: E_P(j) = Σ over holdings of (holding value × exposure of holding to factor j); Exposures are additive across positions, which is why the hedge equations are linear.
Bucket DV01 (key rate DV01): DV01_k = value gain for a 1bp fall in the bucket k yield; Positive for a long bond position. Use a consistent sign convention in the whole question.
P&L from bucket yield changes: P&L ≈ −Σ DV01_k × Δy_k (Δy in bp); Use this form when DV01 is defined as the gain for a 1bp fall. A rise in yield gives a loss for a long position.
Single-factor hedge notional: Hedge notional = Portfolio DV01 ÷ DV01 per unit of hedge instrument; Matches total DV01 only. Leftover bucket DV01s are the residual risk.
Multi-factor hedge: Σ_j (N_j × DV01_j,k) = −Portfolio DV01_k for each bucket k; One equation per bucket. With two buckets and two instruments, solve the two equations together.
Hedge effectiveness: HE = 1 − Var(hedged P&L) ÷ Var(unhedged P&L); Closer to 1 means more risk removed. It depends on the curve moves you test.
DV01-neutral curve trade: N_short-end × DV01_short-end = N_long-end × DV01_long-end; Long and short legs have equal DV01 so that a parallel shift nets to about zero.
Curve trade P&L: P&L ≈ DV01 of each leg × (change in long-end yield − change in short-end yield), with sign by position; Steepener: gain if the spread (long yield − short yield) widens. Flattener: gain if it narrows.

Quick revision

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.

Common mistakes

Using Macaulay duration directly in the price change formula. Fix: Divide Macaulay duration by (1 + y/m) first. Only modified duration gives the price change per unit yield.
Entering basis points as whole numbers, for example Δy = 50 instead of 0.005. Fix: Convert immediately: bp ÷ 10,000. Write the decimal on your paper.
Treating key rate duration as the same as effective duration. Fix: Effective duration is for a parallel shift. A KRD is for one key rate only. Only the sum of KRDs equals effective duration.
Shifting all rates when computing one key rate exposure. Fix: Move only the chosen key rate, with linear interpolation to its neighbours, and leave the other key rates unchanged.
Dividing an eigenvalue by the largest eigenvalue instead of the sum of all eigenvalues. Fix: The denominator is always the total variance, the sum of all eigenvalues.
Treating the eigenvalue as a standard deviation. Fix: Take √λ to get the factor volatility.
Hedging with one instrument per portfolio DV01 and calling it done. Fix: Count the factors. If the question lists exposures at several key rates or principal components, you need one instrument per factor.
Ignoring an instrument's exposure to the other factors. Fix: Include every nonzero exposure in each equation. Solve the longest-end instrument first, then adjust the short-end one.
Treating a total-DV01 hedge as protection against any curve move. Fix: Remember that a single-factor hedge only covers parallel shifts. Always check each bucket's net DV01 after the hedge.
Mixing up steepener and flattener. Fix: Steepener: long the short-end bond, short the long-end bond, profit if the spread widens. Flattener is the reverse. Anchor it to the spread, not to the direction of the level.

Exam tips

Check whether the question gives Macaulay or modified duration before doing anything else.
Expect questions that ask why a DV01 hedge fails under a curve twist. The answer is the parallel shift assumption.
Use DV01 to compare positions. Portfolio DV01 is additive, so you can add positions directly.
If an option sign looks wrong (a price rise when yields rise for a plain bond), discard it.
Convexity corrections are small for small yield moves. Do not spend time on them unless asked.
Questions often give KR01s and a non-parallel move and ask for the value change. Do the dot product and watch the sign.
Check units first. Many wrong options come from using KRD as if it were KR01.
Use the sum rule to eliminate options: the key rate figures must add up to the total DV01 or duration.