📚 Nelson-Siegel-Svannson algorithm 📚
Popular algorithm for fitting a yield curve to obseved 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 Matlab's fminsearch function). 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:
- Observed yield rates
- Maturity of each observed yield
- Initial guess for parameters
- Target maturities
- 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%.
TimeVec = [1; 2; 5; 10; 25]; YieldVec = [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 = [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. disp("Optimal parameters are") disp(OptiParam) ResultYield = NelsonSiegelSvansson(TimeResultVec, OptiParam(1), OptiParam(2), OptiParam(3), OptiParam(4), OptiParam(5), OptiParam(6)); % Calculate the yield for tergeted maturities using the calibrated partameters
Gregor Fabjan (2023). Nelson Siegel Svansson (https://github.com/qnity/nelson_siegel_svansson_matlab), GitHub. Retrieved .
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