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1 Computational Optimization Methods in Statistics, Econometrics and WORKING PAPERS SERIESWPS-031 30/03/2010 Calibrating the Nelson Siegel Svensson modelM. GilliStefan Gro eE. Schumann- Marie Curie Research and Training Network funded by the EU Commission through MRTN-CT-2006-034270 -Calibrating the Nelson Siegel Svensson modelManfred Gilli , Stefan Gro e and Enrico Schumann March30,2010 AbstractThe Nelson Siegel Svensson model is widely-used for modelling the yield curve, yetmany authors have reported numerical difficulties when calibrating the model.

2 We ar-gue that the problem is twofold: firstly, the optimisation problem is not convex and hasmultiple local optima. Hence standard methods that are readily available in statisticalpackages are not appropriate. We implement and test an optimisation heuristic, Differ-ential Evolution, and show that it is capable of reliably solving the model. Secondly, wealso stress that in certain ranges of the parameters, the model is badly conditioned, thusestimated parameters are unstable given small perturbations of the data. We discuss towhat extent these difficulties affect applications of the model of Nelson and Siegel (1987) and its extension by Svensson (1994) are widelyused by central banks and other market participants as a model for the term structure ofinterest rates (Gimenoa and Nave,2009; BIS,2005).

3 Academic studies have provided evi-dence that the model can also be a valuable tool for forecasting the term structure, see forinstance Diebold and Li (2006). Model calibration, ie, obtaining parameter values such thatmodel yields accord with market yields, is difficult; many authors have reported numericaldifficulties when WORKING with the model (for instance, Bolder and Str liski,1999; Gurkay-nak et al.,2006; De Pooter,2007) . In this paper we analyse the calibration of the modelin more detail. We argue that the problem is twofold: firstly, the optimisation problemis not convex and has multiple local optima.

4 Hence methods that are readily available instatistical packages in particular methods based on derivatives of the objective function are not appropriate to obtain parameter values. We implement and test an optimisationheuristic, Differential Evolution, to obtain find that Differential Evolutiongives solutions that fit the data very well. Secondly, we also stress that in certain rangesof the parameters, the model is badly conditioned, thus estimated parameters are unstablegiven small perturbations of the data.

5 We discuss to what extent these difficulties affectapplications of the model. University of Geneva, Switzerland. NordLB, Hanover, Germany. Corresponding M. Gilli and E. Schumann gratefully acknowledge financial support from theeuCommissionthroughmrtn-ct-2006-03427 0 paper is structured as follows: Section2introduces the Nelson Siegel and Nelson Siegel Svensson models and presents an estimation experiment. Section3explains thecollinearity problem; Section4compares results for alternative estimation techniques. and estimationWe look into the two main variants of the model, namely the original formulation of Nelsonand Siegel (1987), and the extension of Svensson (1994).

6 De Pooter (2007) gives an overviewof other and Siegel (1987) suggested to model the yield curve at a point in time as follows:lety( )be the zero rate for maturity , theny( ) = 1+ 2 1 exp( / ) / + 3 1 exp( / ) / exp( / ) .(1)Thus, for given a given cross-section of yields, we need to estimate four parameters: 1, 2, 3, and . Formobserved yields with different maturities 1, .. , m, we is a simple strategy to obtain parameters for this model: fix a , and then estimatethe -values with Least Squares (Nelson and Siegel,1987, ); see also below.

7 We donot assume that the model s parameters are constant, but they can change over time. Tosimplify notation, we do not add subscripts for the time the Nelson Siegel (ns) model, the yieldyfor a particular maturity is hence the sumof several components. 1is independent of time to maturity, and so it is often interpretedas the long-run yield level. 2is weighted by a function of time to maturity. This functionis unity for =0 and exponentially decays to zero as grows, hence the influence of 2is only felt at the short end of the curve.

8 3is also weighted by a function of , butthis function is zero for =0, increases, and then decreases back to zero as grows. Itthus adds a hump to the curve. The parameter affects the weight functions for 2and 3; in particular does it determine the position of the hump. Anexample is shown inFigures1to3. The parameters of the model thus have, to some extent, a direct (observable)interpretation, which brings about the constraints 1>0 , 1+ 2>0 .We also need to have > Nelson Siegel Svensson (nss) model adds a second hump term (see Figure3) tothensmodel.

9 Let againy( )be the zero rate for maturity , theny( ) = 1+ 2 1 exp( / 1) / 1 +(2) 3 1 exp( / 1) / 1 exp( / 1) + 4 1 exp( / 2) / 2 exp( / 2) .Here we need to estimate six parameters: 1, 2, 3, 4, 1and 2. The constraints remain20510024componentyield in %0510024resulting yield curveFigure1: Level. The left panel showsy( ) = 1=3. The right panel shows the corre-sponding yield curve, in this case alsoy( ) = 1=3. The influence of 1is constant for all .0510 202componentyield in %0510024resulting yield curveFigure2: Short-end shift.

10 The left panel showsy( ) = 2 1 exp( / ) / for 2= right panel shows the yield curve resulting from the effects of 1and 2, ie,y( ) = 1+ 2 1 exp( / ) / for 1=3, 2= 2. The short-end is shifted down by2%, butthen curve grows back to the long-run level of3%.0510 202componentyield in %0510024resulting yield curveFigure3: Hump. The left panel shows 3 1 exp( / ) / exp( / ) for 3=6. Theright panel shows the yield curve resulting from all three components. In all panels, same, but we also have 1,2>0. Like forns, we could fix the -values ie, use a gridof different values , and then run a Least Squares algorithmto obtain parameter estimates,even though we cannot handle inequality constraints with a standard Least Squares generally, the parameters of the models can be estimatedby minimising the dif-ference between the model ratesy, and observed ratesyMwhere the superscript stands for market.