Popular algorithm for fitting a yield curve to observed data.
Data on bond yields is usualy avalible only for a small set of maturities, while the user is normaly interested in a wider range of yields.
A popular solution is to use an algorithm to find a function that fits the existing datapoints. This way, the function can be used to interpolate/extrapolate any other point. The Nelson-Siegel-Svannson model is a curve-fitting-algorithm that is flexible enough to approximate most real world applications.
The Nelson-Siegel-Svensson is an extension of the 4-parameter Nelson-Siegel method to 6 parameters. The Scennson introduced two extra parameters to better fit the variety of shapes of either the instantaneous forward rate or yield curves that are observed in practice.
It produces a smooth and well behaved forward rate curve.
The intuitive explanation of the parameters.
beta0is the long term interest rate and
beta0+beta1is the instantaneous short-term rate.
To find the optimal value of the parameters, the Nelder-Mead simplex algorithm is used (Already implemented in the scipy package). The link to the optimization algorithm is Gao, F. and Han, L. Implementing the Nelder-Mead simplex algorithm with adaptive parameters. 2012. Computational Optimization and Applications. 51:1, pp. 259-277.
The furmula for the yield curve (Value of the yield for a maturity at time ‘t’) is given by the formula:
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Observed yield rates
Maturity of each observed yield
Initial guess for parameters
Calculated yield rates for maturities of interest
The user is interested in the projected yield for government bonds with a maturity in 1,2,5,10,25,30, and 31 years. They have data on government bonds maturing in 1, 2, 5, 10, and 25 years. The calculated yield for those bonds are 0.39%, 0.61%, 1.66%, 2.58%, and 3.32%.
from nelsonsiegelsvensson import * import numpy as np TimeVec = np.array([1,2,5,10,25]) YieldVec = np.array([0.0039, 0.0061, 0.0166, 0.0258, 0.0332]) beta0 = 0.1 # initial guess beta1 = 0.1 # initial guess beta2 = 0.1 # initial guess beta3 = 0.1 # initial guess lambda0 = 1 # initial guess lambda1 = 1 # initial guess TimeResultVec = np.array([1,2,5,10,25,30,31]) # Maturities for yields that we are interested in ## Implementation OptiParam = NSSMinimize(beta0, beta1, beta2, beta3, lambda0, lambda1, TimeVec, YieldVec) # The Nelder-Mead simplex algorithem is used to find the parameters that result in a curve with the minimum residuals compared to the market data. # Print the yield curve with optimal parameter to compare with the data provided print(NelsonSiegelSvensson(TimeResultVec, OptiParam, OptiParam, OptiParam, OptiParam, OptiParam, OptiParam))