NelsonSiegel

The Nelson-Siegel model is a parametric representation of the term structure of interest rates. Introduced by Charles R. Nelson and Andrew F. Siegel in 1987, it describes the yield curve with three interpretable components: level, slope, and curvature. The model expresses the yield y(t) for a maturity t as a linear combination of two decaying factors plus a constant: y(t) = β0 + β1 * (1 − exp(−t/λ)) / (t/λ) + β2 * [ (1 − exp(−t/λ)) / (t/λ) − exp(−t/λ) ], where λ is a decay parameter and β0, β1, β2 are loadings associated with the level, slope, and curvature, respectively.

Interpretation and properties: β0 represents the long-term level of interest rates, β1 controls the short-maturity slope of the curve, and β2 captures curvature that can create a hump at intermediate maturities. As maturity grows without bound, y(t) approaches β0, reflecting the model’s focus on the asymptotic level.

Estimation and extensions: The parameters β0, β1, β2, and λ are estimated from observed yields across maturities, typically by nonlinear least squares or Kalman-filter methods for time-series data. Dynamic versions allow the loadings to evolve over time to track shifting curve shapes. The Svensson extension adds a second exponential term to improve fit, particularly for more complex curvature patterns.

Applications and usage: The Nelson-Siegel form is widely used for yield-curve fitting, risk management, and macroeconomic analysis due to its parsimony, interpretability, and computational efficiency. It remains a foundational tool in central banks, financial institutions, and academic research.