How are partial durations (key rate durations) computed and used for targeted hedging?

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AcadiFi · Accepted Answer

Partial duration, or key rate duration (KRD), at maturity t is the sensitivity of the bond's (or portfolio's) price to a 1bp shift in the t-year zero rate (or par rate) while all other key rates remain fixed. The shift is typically modeled as a 'tent' function: full impact at t, declining linearly to zero at neighboring key points. **Computation (numerical)**: 1. Bootstrap the zero curve from current market instruments. 2. For key rate t, shift the zero at t by +1bp and interpolate the rest. 3. Reprice the bond under the shifted curve. 4. Repeat for −1bp shift. 5. KRD_t = [P(−1bp) − P(+1bp)] / (2 × P × 0.0001) **Tent example**: To shift the 10yr KRD, tilt the curve so the 10yr is +1bp, with linear taper to 0 at 5yr and at 30yr. Points outside the tent are unaffected. **Properties**: - Σ KRDs ≈ Effective Duration (for small parallel shifts, tent functions sum to uniform shift) - For bullet bonds (single maturity), KRD is concentrated at the maturity date - For coupon bonds, KRDs spread across coupon dates — high coupons mean more short-end KRD - Barbells (short + long maturity holdings) have split KRDs at the two ends **Barbell hedging example**: Suppose you hold a barbell with: - $60M of 2yr notes: contributes KRD_2y = 1.1, KRD_10y = 0 - $40M of 30yr bonds: contributes KRD_2y = 0, KRD_30y = 8.5 - Portfolio KRD_2y = 1.1 × 0.6 = 0.66 - Portfolio KRD_30y = 8.5 × 0.4 = 3.40 - Parallel duration ≈ 0.66 + 3.40 = 4.06 A bullet 10yr portfolio with the same parallel duration would have KRDs: KRD_10y ≈ 4.0, others 0. Vastly different curve exposure. **Hedging**: To hedge curve risk, you need at least as many hedge instruments as KRDs you care about. For 4 key rates (2, 5, 10, 30), you need 4 hedges — typically Treasury futures at different tenors. Solve a linear system to find hedge ratios that zero out each KRD. ```mermaid flowchart LR A[Portfolio
KRDs] --> B[Hedge Matrix] C[Futures
KRDs] --> B B --> D[Linear Solve
for Hedge Ratios] D --> E[Zero Out
Each KRD] ``` **Practical use**: - **Immunization**: Match portfolio KRDs to liability KRDs for full curve-matched immunization (vs simple duration matching which leaves curve residual). - **Active views**: Express a flattener (long 30yr / short 2yr) by targeting negative KRD_2y and positive KRD_30y with offsetting parallel duration. - **Risk reports**: KRD tables reveal where your curve bets are concentrated — a seemingly duration-neutral book can have large off-setting curve bets. KRDs are the workhorse framework for any fixed-income PM past beginner level — they transform 'duration' from a single number into a rich picture of curve exposure, enabling both targeted hedging and more precise attribution of P&L.

How are partial durations (key rate durations) computed and used for targeted hedging?

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