Parametric Term Structure Federal Reserve / ECB Model

Nelson-Siegel & Svensson Yield Curve Fitting Lab

Calibrate institutional parametric yield curves using Nelson-Siegel (1987) and Svensson (1994) models. Decompose the term structure into Level ($\beta_0$), Slope ($\beta_1$), and Curvature ($\beta_2, \beta_3$) components, derive instantaneous forward curves, and stress-test macroeconomic shifts.

Empirical Calibration Presets:
Model Architecture & Beta Parameters
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Hump 1 peak: 1.33Y
Hump 2 peak: 5.00Y
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Short Rate (t=0)

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y(0) = β0 + β1
Long Rate (t=∞)

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y(∞) = β0
10Y - 2Y Slope

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Steepness spread
Fit Quality (RMSE)

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Excellent Fit
Yield Curve & Loading Dynamics

Continuous Zero-Coupon Spot Curve $y(t)$ vs. Instantaneous Forward Curve $f(t)$ overlaid with discrete market benchmark yields (2Y, 5Y, 10Y).

Parametric basis loadings across maturity $t$: Level (constant 1.0), Slope $\frac{1-e^{-\lambda t}}{\lambda t}$, and Curvature hump functions.

Decomposition of total spot yield into Level contribution, Slope contribution, and Curvature contributions across tenors.

Parametric Yield Curve Schedule & Fitting Residuals
Maturity Tenor Model Spot Yield ($y_t$) Instant Forward ($f_t$) Discount Factor ($P_t$) Market Benchmark Model Residual (bps)
Sensitivity Matrix 1: Level ($\beta_0$) vs. Slope ($\beta_1$) on 10Y Yield (%)
Rows: Level Factor $\beta_0$ (%) • Columns: Slope Factor $\beta_1$ (%). Evaluates how macro level and steepness jointly drive intermediate/long benchmark rates.
Sensitivity Matrix 2: Decay ($\lambda_1$) vs. Curvature ($\beta_2$) on 2Y Yield (%)
Rows: Decay Parameter $\lambda_1$ • Columns: Curvature Factor $\beta_2$ (%). Demonstrates how decay speed shifts the belly of the curve.

Nelson-Siegel & Term Structure Mastery Quiz

1. In the Nelson-Siegel model, what does the instantaneous short rate $y(0)$ equal as maturity $t \to 0$?

2. At what maturity tenor does the first curvature hump loading reach its theoretical maximum?

3. Why did Lars Svensson introduce a second curvature factor $\beta_3$ and second decay parameter $\lambda_2$?

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Frequently Asked Questions: Nelson-Siegel & Svensson Curve Models

The Nelson-Siegel model is a parsimonious parametric formula that represents the zero-coupon yield curve as a combination of three economic components: a long-term level factor (β0), a short-term slope factor (β1), and a medium-term curvature or hump factor (β2), modulated by a scale/decay parameter (λ). It guarantees smooth, continuous yield and forward curves across all maturities.

Lars Svensson expanded the 4-parameter Nelson-Siegel model into a 6-parameter model by adding a second curvature factor (β3) with an independent decay parameter (λ2). This allows the curve to capture complex shapes, such as dual-humped or S-shaped term structures often observed during monetary policy transition regimes.

β0 represents the asymptotic long-term interest rate as maturity approaches infinity (level). β1 represents the spread between short-term rates and the long-term rate (as t approaches 0, y(0) = β0 + β1; thus -β1 measures the curve slope). β2 measures the magnitude of the medium-term hump or trough (curvature), peaking around maturity 1/λ.

While cubic or smoothing splines can overfit local pricing anomalies or trade-specific liquidity noise, the NSS model enforces global economic structural smoothness and sensible asymptotic behavior at the ultra-long end, preventing negative or exploding forward rates.

By applying the continuous forward relationship f(t) = y(t) + t × [dy(t)/dt], the Nelson-Siegel formula yields a remarkably simple closed-form forward curve: f(t) = β0 + β1 × e^(-λt) + β2 × (λt) × e^(-λt). In the Svensson model, the second hump term adds + β3 × (λ2·t) × e^(-λ2·t).