Applied Mathematical Sciences, Vol. 9, 2015, no. 25, 1201 - 1212

HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.43209

Yield Curve Estimation of the Nelson-Siegel Class

Model by Using Hybrid Method with L-BFGS-B

Iterations Approach

Muslim

Gadjah Mada University, Yogyakarta, Indonesia

Department of Mathematics Education

Jambi University, Jambi, Indonesia

Dedi Rosadi

Department of Mathematics

Gadjah Mada University, Yogyakarta, Indonesia

Gunardi

Department of Mathematics

Gadjah Mada University, Yogyakarta, Indonesia

Abdurakhman

Department of Mathematics

Gadjah Mada University, Yogyakarta, Indonesia Copyright © 2014 Muslim et al. This is an open access article distributed under the Creative

Commons Attribution License, which permits unrestricted use, distribution, and reproduction in

any medium, provided the original work is properly cited.

Abstract

This paper discussed about model extension in determines the yield curve.

Determine of yield curve using Nelson-Siegel class model. This class model

consisting of: 3-factor model, 4-factor model, the 5-factor model, and 6-factor

model. 6-factor model is a model extended from 5-factor models. The extension

aims to increase the level of accuracy in determine the yield curve. Nelson-Siegel

class model is model that more difficult to estimate because it has two shape the

1202 Muslim et al.

parameters, i.e. the linear and nonlinear parameters. Extension of this model is

done by adding the fourth hump into 5-factor model. In addition, we obtain new

model, this model have local minimum multiple so that it is more difficult to be

estimated. To estimate this model, we propose estimation using a hybrid method.

Hybrid method is combines method of estimation the nonlinear least squares with

constrained optimization, and then continued with L-BFGS-B iteration approach.

Estimation of the class model was done by full estimation, i.e. estimating the linear

parameters and nonlinear parameters simultaneously. Then, we calculated MSE,

AIC, and BIC. The purpose of calculating this component is to determine the best

of model. The best model obtainable if the models have component value which is

smaller than the other models. This paper uses data from Indonesian government

bonds. Based on data processing, we obtained the best model i.e. 6- factors model.

Keywords: Linear parameter, nonlinear parameter, constraint optimization, NLS,

L-BFGS-B, and the best of model

1 Introduction

In finance, we know yield curve. The curve is a curve that describes yield to

maturity of bonds. Based on given of interest rate, the curve consists of three parts

i.e. zero coupon, forward rate, and discount factor. To determine these curves can

be used various model, one of the model that using is Nelson-Siegel class model.

This class model, namely: Nelson-Siegel (3-factor) model, Svensson (4-factor)

model, Ferreira-Rezende (5-factor) model, and extension of Nelson-Siegel class

model by adding the second minimum curve into 4-factor model so that we obtain

5-factor new model.

Papers that discuss Nelson-Siegel class model are Svenson [25], Bolder and

Strzelecki [2], Brousseau [3], Jankowitsch and Pichler [16], Diebold et al. [10],

Diebold and Li [7], Diebold et al [8], Diebold et al[11], Diebold et al [9], Krippner

[19], Ejsing et al[12], Diebold et al [6], Bauer [1], Cristensen et al [4], Christensen

et al [5], Krippner [19], Ferstl and Hayden [13], Gilli et all [14], Rezende and

Ferreira [23], and Rosadi [24].

3-factor models are a model that contains flat curve, hump, and S shape

(Nelson-Siegel, [21]). This model was expanded again by adding the second humps

into 3-factor model (Svensson, [25]). It is known as Nelson-Siegel and Svensson

model or 4-factor model. Rezende and Ferreira [23] expand again by adding second

minimum curve into 4-factor model so this model has two the minimum curve. This

model is Ferreira- Rezende model or better known as 5-factor model. In the class

model, there are still disadvantages so it is necessary for the extension in obtain

accuracy the yield curve. The expansion of this model is done by adding the third

hump into 4-factor model. The expansion causes 4-factor model become 5-factor

new model.

Increase of humps into NS class model; make this class model more difficult to

estimate. The difficulty caused this class model has local minimum multiple, also

Yield curve estimation of the Nelson-Siegel class model 1203

because the class model has linear and nonlinear parameters. The class model has

minimum local multiple and it has two shape parameters so that the class model

can’t be estimated using ordinary estimation. Overcome this problem, we use

estimation of hybrid method. This estimation method uses hybrid between

estimation and optimization. In hybrid the estimation and optimization must to have

relationships that are interrelated so that hybrid can be solving for determination

each parameter, both linear and nonlinear. Estimates used in the hybrid method i.e.

Nonlinear Least Squares (NLS) while optimization is used constraint optimization

then we perform iterations using L-BFGS-B method. A description of NLS can be

seen in Sun and Yuan [27] and Björck [28].

Constraint optimization is optimization that consists of constraints multiple;

explanation of optimization can be seen in Nocedal and Wright [28], Griva, et.al

[15], and Rao [29]. Iteration of L-BFGS-B method is iteration extended of L-BFGS

iteration that handles of simple bounds has this model (Zhu et al. [26]. The steps

using hybrid method as follows: (1) determine the sum square error (SSE) of each

model, (2) The SSE is estimated using NLS, (3) The last estimate in optimization

using L-BFGS-B. A description of the steps can be seen in section (5).

2 Nelson-Siegel Class Model

In this paper, determine the yield curve using Nelson-Siegel class model. This

class model is consisting of linear and nonlinear parameters. In this class model is

found some models, i.e., 3-factor model, 4-factor model, 5-factor model, and

6-factor model. Explanation of this class model, as follows:

2.1 3-factor Model

This model was discussed by Charles Nelson and Andrew Siegel of the

University of Washington in 1987 (Bolder and Stréliski, [2]). The model curve

consists of flat shape, hump, and S shape (Nelson and Siegel, [21]). This model is

formulated as follows:

𝑦𝑖(𝜆; 𝜷, 𝜏) = 𝛽0 + 𝛽1 𝑒𝑥𝑝 (−𝜆𝑖

𝜏) + 𝛽2

𝜆𝑖

𝜏exp (−

𝜆𝑖

𝜏) , (1)

where 𝑦 is yield, 𝜆 is time to maturity, 𝜷 is linear parameter i.e. 𝜷 =(𝛽0, 𝛽1, 𝛽2)′ , 𝜏 is nonlinear parameter, and 𝑖 is index of bond. 𝛽0 is earlier

function of time to maturity, 𝛽1 determine the recovery curve (short-term) in

various irregularities, This curve will be negatively skewed if the positive

parameters, forward rate function will converge to 𝛽0 + 𝛽1 if 𝜏 approaching

infinity, 𝛽2 determine the magnitude and direction of maximum, and 𝜏

determine the maximum position or shape of the curve U. The parameters of this

model has bounds i.e. 𝛽0 > 0, 𝛽0 + 𝛽1 > 0, and 𝜏 > 0.

1204 Muslim et al.

2.2 4-factor Model

Extension of the 3-factor model is proposed by Lars E. O. Svensson in 1994.

The expansion is done by adding second hump in the 3-factor model. This

resulted in the addition of 3-factor models become 4-factor model. This model is

called the Svensson model, function of the model is formulated as follows:

𝑦𝑖(𝜆; �̇�, 𝝉) = 𝛽0 + 𝛽1 𝑒𝑥𝑝 (−

𝜆𝑖

𝜏1) + 𝛽2 [

𝜆𝑖

𝜏1exp (−

𝜆𝑖

𝜏1)] + 𝛽3 [

𝜆𝑖

𝜏2exp (−

𝜆𝑖

𝜏2)] , (2)

where 𝑦 is yield, 𝜆 is time to maturity, �̇� is linear parameters i.e. �̇� =(𝛽0, 𝛽1, 𝛽2, 𝛽3)′, 𝝉 is nonlinear parameters by 𝝉 = (𝜏1, 𝜏2)′, and 𝑖 is index of bond

by 𝑖 = 1, 2, 3, ⋯ , 𝑝. Term of 𝛽0 is constant value of forward rate functions, it is

always constant if term maturity approaches zero, 𝛽1 determine starting values

curve (short-term) in various forms of deviance, curve will be negatively skewed

if the parameter is positive and vice versa, 𝛽2 is determine the magnitude and

direction of the maximum curve, if 𝛽2 positive then maximum will occur in 𝜏1,

if 𝛽2 negative then it will form U curve on 𝜏1 and 𝛽3 equal 𝛽2 which

determines both magnitude and direction of maximum, 𝜏1 determine special

position of the first maximum or U shape curve, 𝜏2 determine second maximum

position or shape of the curve U. The parameters of this model also has bounds i.e.

𝛽0 > 0, 𝛽0 + 𝛽1 > 0, and 𝜏1,2 > 0.

2.3 5-factor Model

The model developed by Rezende and Ferreira (Rezende and Ferreira, [23]).

Developed of the model is addition the second minimum curve in Svensson model.

This model is often called 5-factor model or Ferreira-Rezende model. In

mathematically, the model is written as follows:

𝑦𝑖(𝜆; �̈�, �̇�) = 𝛽0 + 𝛽1 𝑒𝑥𝑝 (−𝜆𝑖

𝜏1) + 𝛽2 𝑒𝑥𝑝 (−

𝜆𝑖

𝜏2)

+𝛽3 [𝜆𝑖

𝜏1exp (−

𝜆𝑖

𝜏1)] + 𝛽4 [

𝜆𝑖

𝜏2exp (−

𝜆𝑖

𝜏2)], (3)

𝑦 is yield, 𝜆 is time to maturity, �̈� is linear parameters by �̈� =(𝛽0, 𝛽1, 𝛽2, 𝛽3, 𝛽4)′, �̇� is nonlinear parameters by �̇� = (𝜏1, 𝜏2)′, and 𝑖 is index o f

time to maturity. The Bounds of parameters are 𝛽0 > 0, 𝛽0 + 𝛽1 > 0, and 𝜏12 >0.

Yield curve estimation of the Nelson-Siegel class model 1205

2.4 5-factor new Model

This model, we developed from previous models. This development aims to

increase the level of accuracy of the model. In determining the yield curve, the

extension is done by adding the third minimum curve into the 4-factor model. The

addition is focused expansion the 4-factor model. Extension of this model is

formulated as follows:

𝑦𝑖(𝜆; 𝜃) = 𝛽0 + 𝛽1 𝑒𝑥𝑝 (−𝜆𝑖

𝜏1) + 𝛽2 [

𝜆𝑖

𝜏1exp (−

𝜆𝑖

𝜏1)]

+𝛽3 [𝜆𝑖

𝜏2exp (−

𝜆𝑖

𝜏2)] + 𝛽4 [

𝜆𝑖

𝜏3exp (−

𝜆𝑖

𝜏3)] (4)

where 𝑦 is yield, 𝜆 is time to maturity, �⃛� is linear parameters, i.e. �⃛� =(𝛽0, 𝛽1, 𝛽2, 𝛽3, 𝛽4, 𝛽5)′ , �̈� is nonlinear parameters, i.e. �̈� = (𝜏1, 𝜏2, 𝜏3)′ , 𝜽 =

(�⃛�, �̈�)′and 𝑖 is index of bond by 𝑖 = 1, 2, 3, ⋯ , 𝑝, 𝛽0 is constant value of forward

rate function, 𝛽1 determine curve earlier value (short term) in various of

magnitude term, curve will be negatively skewed if 𝛽1 positive, vice versa, 𝛽2

determine of shape first maximum curve, this curve will be positively skewed if 𝛽2

positive and negatively skewed if 𝛽2 negative, 𝛽3 determine of shape second

maximum curve, this curve will be positively skewed if 𝛽3 positive and will be

shape S if 𝛽3 negative. The next, 𝛽4 determine of shape third maximum curve, this

curve will be positively skewed if 𝛽4 positive and shape S if 𝛽4 negative, 𝛽5

determine of shape fourth maximum curve, this curve will always positive if 𝛽5

postive or negative, 𝜏1 determine position of first maximum curve, 𝜏2 determine

position of second maximum curve, 𝜏3 determine position of third maximum

curve, and 𝜏4 determine position of fourth maximum curve. This model is

illustrated in the following Figure 1:

Figure 1. Yield Curve Component of 6-Factors Model

1206 Muslim et al.

Figure 1 shows that the flat curve is the first curve where the value is always

constant, the minimum curve is the curve of the second factor where it will be

negatively skewed if 𝛽1 is positive and vice versa. First hump is the third factor

of Nelson-Siegel model, second hump is the additional factor proposed by

Svensson (1994), third hump is an expansion in which is done by adding the curve

into 4-factors model proposed by Rezende-Ferriera, and the last is fourth hump

that is additional curve we propose here to 5-factors model extended and bounds

the parameters is 𝛽0 > 0, 𝛽0 + 𝛽1 > 0, and 𝜏1,2,3 > 0.

3 Estimation Methods

To determining parameters of the Nelson-Siegel class model, we estimate the

parameters as such as to minimize the sum of squared errors between the estimated

yields, 𝑦𝑆𝐹, and observed yields to maturity ,𝑦, so, we can write as follows

𝜃𝑖 = arg min𝜃𝑖

∑ (𝑦𝑖𝑆𝐹 − 𝑦𝑖)

2𝑝𝑖=1 , (5)

The sum of squared errors in equation (5) using nonlinear least square, this equation

has inequality constraints, the constraints containing some parameters in each

model. These constraints can be transformed to equality constraint by adding the

slack variable a non-negative (Rao, [29]). Let 𝑐𝑗 > 0 and 𝑗 = 1,2, ⋯ , 𝑛 is

inequality constraints of Nelson-Siegel class model, this constraint transformed

into equality constraint, i.e. 𝑐𝑗(𝜃) − 𝑎𝑗2 = 0, the equation (5) can be optimized by

equality constraints, as follows:

𝜃𝑖 = arg min𝜃𝑖

∑ (𝑦𝑖𝑆𝐹 − 𝑦𝑖)

2𝑝𝑖=1 , (6)

subject to

𝑐𝑗(𝜃) − 𝑎𝑗2 = 0,

Optimization of equation (6) subject to equality constraints can be formed Lagrange

function, as follows

ℒ(𝜃, 𝝁) = 𝜃𝑖 − ∑ 𝜇𝑗

𝑛

𝑗

𝑐𝑗(𝜽) − 𝑎𝑗2, (7)

where 𝜃 is parameters of function, 𝝁 = (𝜇1, 𝜇2, ⋯ , 𝜇𝑛)′ is Lagrange vector with

non-positive, �̂�𝑖 is the sum of squared errors of each models, c is constraint of each

models, and 𝑎 = (𝑎1, 𝑎2, ⋯ , 𝑎𝑛)′ is vector of slack variable. Explanation more

details can be seen Rao [29].

To minimize equation (7), we use L-BFGS-B iteration method approach. This

iteration method is an extension of the method L-BFGS or LM-BFGS (limited

memory BFGS) which has parameters bounded. BFGS method is combination

Yield curve estimation of the Nelson-Siegel class model 1207

method proposed by Broyden, Fletcher, Goldfarb, and Shanno. This method is

extension of quasi-Newton method or the descent method (Griva et.al, [15]). As

presented by Kelly [17].

Given Lagrange functions, as follows:

min ℒ(𝜃, 𝝁), 𝜃, 𝝁 ∈ Ω (8) where

Γ ∈ Ω = {𝜃 ∈ ℜN|𝑙𝑘 ≤ 𝜃𝑘 ≤ 𝑢𝑘}

𝑙𝑘 and 𝑢𝑘 each is the upper and lower bound parameters of 𝜃𝑘, 𝜃 is parameters

of Nelson-Siegel model class. The magnitude of above parameters calculated by

BFGS method, as formula:

𝐵𝑘+1 = 𝐵𝑘 + (1 +𝑔𝑘

𝑇𝐵𝑘𝑔𝑘

𝑑𝑘𝑇𝑔𝑘

)𝑑𝑘𝑑𝑘

𝑇

𝑑𝑘𝑇𝑔𝑘

−𝑑𝑘𝑔𝑘

𝑇𝐵𝑘

𝑑𝑘𝑇𝑔𝑘

−𝐵𝑘𝑔𝑘𝑑𝑘

𝑇

𝑑𝑘𝑇𝑔𝑘

. (9)

where 𝐵𝑘 is approximation inverse of Hessian matrix, 𝐵0 = 𝐼 , 𝜃𝑘+1 = 𝜃𝑘 −𝛿𝑘𝑆𝑘 , and 𝑑𝑘 = 𝜃𝑘+1 − 𝜃𝑘 = 𝛿𝑘𝑆𝑘 , 𝛿𝑘 is distant of interval, 𝑆𝑘 is search

direction, 𝑔𝑘 = ∇ℒ𝑘+1(𝜃) − ∇ℒ𝑘(𝜃) , 𝑆𝑘 = −𝐵𝑘∇ℒ𝑘(𝜃) . Iteration will stop on

‖∇ℒ𝑘(𝜃)‖ ≤ 휀, and 휀 is very small values.

optimization steps can be summarized in estimating the model, namely:

1. Define the sum of squared errors function of each model (equation (1), (2), (3),

and (4);

2. Specify is NLS estimation, i.e. 𝜑 =1

2∑ [𝑟𝑖]

2𝑝𝑖=1 of each parameters subject to

inequality constraints (equation (6));

3. be formed Lagrange function with constraints of objective function (equation

7);

4. specify the value of the matrix 𝐵1 = 𝐼 , 𝑝 × 𝑝 , where 𝑝 is number of

parameters estimated in each model;

5. specify the initial value of the parameter vector with the order parameters 𝑝 ×1, 𝑝 is number of parameters estimated in each model;

6. determine the gradient of the step (2) with each parameters in the model e.g.

∇ℒ𝑖, 7. substitute the initial value of the parameters (step 3) to the gradient of step 5

with the results e.g. ∇ℒ1,

8. specify the value of 𝑔1 by substituting the value of the matrix in step 3 and

initial value of parameter in step 6; 9. specify the value of 𝛿1, so we will obtain 𝑔1and 𝑠1; 10. substitute these values into the formula optimization method L-BFGS-B; 11. Continue this step until the parameter values converge to a value of 0.

1208 Muslim et al.

6 Case Studies

In this section, we will discuss the results of the empirical estimation using the

data of government bonds, the bond data is obtained on May 7, 2010, and this data

can be obtained by visiting http://www.idx.co.id. The data is processed by using

construct programming of algorithm. In running the programming, we choose the

initial value of each model. From these selections obtained results of empirically

estimation. These results, we present in form yield curve followed by the

parameter values of each models. This model classes will be identified to select

the best model, identification using three indicators, namely: the mean square

error (MSE), Akaike’s Information Criterion (AIC), and Bayesian Information

Criterion (BIC). All proceeds from this estimation can be considered as Figure 2:

Figure 2 is yield curve of Nelson-Siegel model class. This figure show spread of

data the yields to maturity. Red curve shows the yield curve of the 3-factor

models, green curve shows the yield curve of the 4-factor and 5-factor models,

and blue curve shows the yield curve of 6-factor model. Yield curve of the

4-factor and 5-factor models shows this curve coincides because these models

have similarity. Parameter values of the models shown in Table 1:

0 5 10 15 20 25

78

91

0

Figure 2. Yield Curve of Nelson-Siegel Model Class

Time to Maturity (year)

Yie

ld to

Ma

turi

ty (

%)

3-factors

4-factors and 5-factors

6-factors

Yield curve estimation of the Nelson-Siegel class model 1209

Table 1. Parameters Values of Nelson-Siegel Model Class

Parameters Models

3-factor 4-factor 5-factor 6-factor

𝛽0 10.7628 10.7099 10.7103 10.5980

𝛽1 -3.6988 -4.0611 -4.0611 -2.8310

𝛽2 -0.00002 -0.9808 -2.2922 -2.5264

𝛽3 - -4.5836 -2.2922 -2.4034

𝛽4 - - -2.4034 -2.4034

𝛽5 - - - 0.3504

𝜏 8.8799 - - -

𝜏1 - 1.6616 1.6609 4.9437

𝜏2 - 4.8540 4.8552 0.2146

𝜏3 - - 4.8552 0.2146

𝜏4 - - - 0.2147

In four models, we select the best model, the best model selected by using

indicators, namely: the mean square error (MSE), Akaike’s Information Criterion

(AIC), and Bayesian Information Criterion (BIC). Based on the estimates used

above, we obtain as table 2:

Table 2. Comparison the Value of MSE, AIC, and BIC

Models MSE AIC BIC

3-factor 0.0716 -1.6092 -2.5589

4-factor 0.0680 -1.6486 -2.5730

5-factor 0.0680 -1.6349 -2.5342

6-factor 0.0559 -1.8211 -2.6951

In Table 2 above shows that 6-factor model have MSE difference of 3-factor,

4-factor, and 5-factor models, respectively are 1.6%, 1.2%, and 1.2%, than AIC

have different are 21%, 17.3%, and 18.6% and BIC have different are 13.6%,

12.2%, and 16.1%. From these result, we conclude that 6-factor model is the best

model.

7 Conclusions

Nelson-Siegel model class is parametric model that has linear and nonlinear

parameters. This model classes are 3-factor model, 4-factor model, 5-factor model,

and 6-factor model. Within this model, estimated parameters are very difficult to

solve, the difficulty is because this model has two types of parameters as

mentioned above. Estimation methods used in the model class, namely: hybrid

method, this method is hybrid between the estimation methods with optimization.

1210 Muslim et al.

Estimates used are nonlinear least square and optimization methods used is

inequality optimization, and continued with iteration of L-BFGS-B method.

Step-by-step use of this estimation method, i.e. each model in the model class

determined SSE function, caused this function is nonlinear, and then be solved by

L-BFGS-B method iteration. The estimation results, then calculated MSE, AIC,

and BIC of each model, calculation results are compared for obtain the best models.

In this selection, we obtain that 6-factor model have indicator smaller than other

models. This optimization has the disadvantage that in determining the initial value

of the parameter, initial value in a parameter is not necessarily appropriate for other

parameters so that we need to specify the initial value for each data.

Acknowledgements. The authors would like to thank the anonymous referee and

the all friends so the manuscript can be compiled into a paper.

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Received: March 15, 2014; Published: February 13, 2015