Popular algorithm for fitting a yield curve to observed data.
Data on bond yields is usually available only for a small set of maturities, while the user is normally 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. 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.
Advantages:
- It produces a smooth and well-behaved forward rate curve.
- The intuitive explanation of the parameters.
beta0is the long-term interest rate andbeta0+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 formula 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
YieldVec. - Maturity of each observed yield
TimeVec. - Initial guess for parameters
beta0,beta1,beta2,beta3,labda0, andlambda1. - Target maturities
TimeResultVec.
- Calculated yield rates for maturities of interest
TimeResultVec.
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 is 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 algorithm 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(NelsonSiegelSvansson(TimeResultVec, OptiParam[0], OptiParam[1], OptiParam[2], OptiParam[3], OptiParam[4], OptiParam[5]))