Nelson-Siegel Yield Curve Calibrator

A handful of observed yields, fit into three interpretable factors: level, slope, and curvature. This is the same parametric backbone central banks actually use to build a smooth curve from scattered market data.

How To Use This Model

Reading This Tool

Enter observed yields at five key maturities.

The tool grid-searches the decay parameter λ and runs OLS regression at each candidate to find the best-fitting level, slope, and curvature factors, then plots the smooth fitted curve against your observed points.

Observed Yields (%)

3-Month Yield 4.80%

2-Year Yield 4.30%

5-Year Yield 4.00%

10-Year Yield 4.15%

30-Year Yield 4.50%

λ is searched over a grid from 0.5 to 10 in steps of 0.1, selecting the value that minimizes total squared fitting error across all five observed points.

Fitted Nelson-Siegel Factors

Level (β₀)

Slope (β₁)

Curvature (β₂)

Decay (λ)

Fitted Yield Curve

Term Structure Econometrics

The Core Formula

y(τ) = β₀ + β₁·[(1−e^−τ/λ)/(τ/λ)] + β₂·[(1−e^−τ/λ)/(τ/λ) − e^−τ/λ]

For any fixed λ, this equation is linear in β₀, β₁, β₂, and can be estimated by ordinary least squares. This tool grid-searches λ and re-runs OLS at each candidate, selecting the λ that minimizes total squared error, a standard, tractable way to handle the model's one genuinely nonlinear parameter.

When To Actually Use This Model

Building a smooth, arbitrage-consistent yield curve from a sparse set of observed bond or swap yields.
Decomposing yield curve movements into level, slope, and curvature shocks for term structure or macro-finance research.
Teaching parametric term structure models in a fixed income or empirical asset pricing course.
Central bank and Treasury applications where a stable, well-behaved functional form is preferred over a fully flexible spline.

Key Assumptions & Limitations

Only three shape factors are estimated; the model cannot fit unusually shaped curves with multiple humps or kinks.
The Svensson extension adds a second curvature term and decay parameter for more flexibility with very rich data.
With only five observed points here, the fit is illustrative, production use typically fits dozens of bond yields simultaneously.
λ determines the maturity at which the curvature term is maximized; grid search here is coarse (0.1 steps) for speed.

Foundational Reference

Nelson, C. R., & Siegel, A. F. (1987). Parsimonious Modeling of Yield Curves. The Journal of Business, 60(4), 473-489.

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