Fixed Income Analytics Ho (1992) Curve Model

Key Rate Duration & Yield Curve Sensitivity Lab

Model non-parallel yield curve reshaping, key rate duration vectors ($KRD_k$), and multi-tenor portfolio sensitivity. Overcome the limitations of single-number effective duration by stress-testing steepening, flattening, and butterfly twists across benchmark tenors (2Y, 5Y, 10Y, 30Y).

Institutional Presets:
Portfolio & Yield Curve Parameters
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Total Allocation: 100%
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Effective Duration

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Σ KRD Vector sum
Front-End KRD (2Y+5Y)

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Short tenor risk
Long-End KRD (10Y+30Y)

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Duration heavy wing
Scenario Price Impact

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Yield Curve Risk Visualizer

Key Rate Duration profile by benchmark node ($KRD_k = -(1/P) \cdot \Delta P_k / \Delta y_k$). The sum of all bars equals the total effective duration.

Baseline par/spot curve vs. scenario-shifted yield curve showing non-parallel steepening/flattening across 2Y, 5Y, 10Y, and 30Y tenors.

Dollar P&L attribution by key rate node: $\text{Dollar P&L}_k \approx -\text{Notional} \cdot KRD_k \cdot \Delta y_k$.

Key Rate Duration Vector & Scenario Attribution Table
Key Rate Tenor Baseline Yield Key Rate Duration (Years) % of Total Duration Scenario Shift Estimated Price Return Dollar Impact ($)
Sensitivity Matrix 1: 2Y vs. 10Y Key Rate Shocks on Portfolio P&L (%)
Rows: 2Y Rate Shift (bps) • Columns: 10Y Rate Shift (bps). Evaluates non-parallel curve slope steepening and flattening impacts.
Sensitivity Matrix 2: 10Y vs. 30Y Key Rate Shocks on Portfolio P&L (%)
Rows: 10Y Rate Shift (bps) • Columns: 30Y Rate Shift (bps). Evaluates long-end curve curvature and twist sensitivity.

Yield Curve & KRD Mastery Self-Assessment

1. What is the fundamental relationship between a bond's individual Key Rate Durations and its Effective Duration?

2. In a "Bear Steepener" scenario where 30-year yields surge +75 bps while 2-year yields only increase +20 bps, which portfolio suffers the greatest loss?

3. How can an asset manager immunize a portfolio against a non-parallel yield curve twist or butterfly shift?

Related Fixed Income & Yield Curve Simulators

Spot vs. Forward Rate Lab Barbell vs. Bullet Lab Bond Convexity Lab Bond Duration Lab Bond Immunization Lab Convertible Bond Lab Interest Rate Swap Lab Credit Spread Lab Value at Risk (VaR) Lab

Frequently Asked Questions: Key Rate Duration & Curve Risk

Key Rate Duration (KRD) measures the price sensitivity of a bond or portfolio to a shift in a single key maturity point along the yield curve, holding all other benchmark key rates constant. Mathematically, KRD_k = -[P(y + Δy_k) - P(y - Δy_k)] / [2 × P_0 × Δy_k], using a finite-difference bump (typically 1 to 10 basis points) distributed via linear triangular tent functions.

Effective Duration assumes that the entire yield curve moves in a strictly parallel fashion (all maturities shift by identical basis points). In real financial markets, empirical yield curve shifts are non-parallel (steepening, flattening, or twists). Key Rate Duration decomposes total duration into a vector across tenors, revealing hidden curve exposure that parallel duration obscures.

Under standard triangular tent interpolation functions where the sum of bump weights equals 1.0 across the entire curve, the sum of all individual Key Rate Durations equals the bond or portfolio's total Effective Duration: Σ KRD_k = D_eff. If all key rates experience an identical parallel shock Δy, the percentage price change is -D_eff × Δy.

Two portfolios with the exact same effective duration (e.g. 7.0 years) can possess radically different KRD vectors. A 10-year bullet portfolio has high KRD at the 10Y tenor and zero at 2Y or 30Y. A barbell portfolio has high KRD at 2Y and 30Y and zero at 10Y. If the curve flattens or twists, the two portfolios experience opposite capital performance despite having identical effective duration.

For any arbitrary vector of key rate shifts [Δy_1, Δy_2, ..., Δy_m], the percentage price change of the bond or portfolio is approximated as: ΔP / P ≈ -Σ (KRD_k × Δy_k) + 0.5 × C × (Δy_avg)². This provides institutional traders with a multi-factor vector Taylor expansion across yield curve points.