APPENDIX D

Parametric and non-parametric yield curve models

D.1 PARAMETRIC CURVE MODELLING

In this case the form of a yield curve is assumed to follow a particular pattern that can be modelled by a mathematical function with a small number of parameters. These parameters can then be supplied in place of the raw curve data.

The best-known parametric model for yield curve modelling is from Nelson and Siegel (1987) who proposed the following expression for a curve’s yield at an arbitrary maturity τ:

• The four parameters β₀, β₁, β₂ and λ define the curve.
• This function matches a wide range of yield curve behaviours, such as sloped, flat, curved, inverted and humped.
• Virtually any curve observed in the marketplace can be fitted to a Nelson-Siegel function, using least-squares or similar techniques.
• The value of β₀ is the asymptotic (or long) yield as τ becomes very large.
• β₀ + β₁ is the short (or spot) rate.
• β₁ represents the difference between the short- and long-term rates, which may be interpreted as the slope of the curve.

D.2 SPLINE MODEL

A spline model typically fits a first or second order polynomial between successive curve data points. There are many refinements on this approach that include considerations such as ensuring the curve is globally smooth.