Abstract. Nonparametric approaches of estimating the yield curve have been widely used as alternative approaches that supplement parametric approaches. In this paper, we propose a novel yield curve estimating algorithm based on radial basis function networks, which is a nonparametric approach. The proposed method is devised to improve accuracy and smoothness of the ﬁtted curve. Numerical experiments are conducted for 57 U.S. Treasury securities with diﬀerent maturities and demonstrate a signiﬁcant performance improvement to reduce test error compared to other existing algorithms.

1

Introduction

Along with stocks and loans, bonds are the main asset class with which the government or a private enterprise can raise money. The term structure of interest rates plays a key role in the reasonable bond-pricing. Because it is based on an interest rate model, the model need be calibrated before using it. The fundamental object to calibrate against is the yield curve of bonds. The ﬁtting techniques of yield curves to yield data or maturity data are divided into two categories, parametric approaches and nonparametric approaches. A popular parametric approach is a family of Nelson-Siegel curves [2]. Despite its simple structure, many parametric approach, however, revealed poor performance in yield curve ﬁtting. As alternative approaches, nonparametric approaches have recently been widely used. Especially, cubic B-splines, which are the kernel functions for estimating a bond yield, are popularly chosen among various nonparametric approaches [3], [5], [13], [14]. In this paper, we propose a novel ﬁtting method for the yield curve. The proposed method comprises two phases. The ﬁrst phase ﬁnds an initial yield curve using a radial basis function network (RBFN) whose inputs are maturities of bonds and outputs are their yields. In the second phase, the initial yield curve predicted by the RBFN in the ﬁrst phase is calibrated to minimize bond-pricing error. The performance of the proposed algorithm is veriﬁed concerned with improving the curve-ﬁtting accuracy and regularization by applying it to notes and bonds issued by U.S. Treasury Department. J. Wang, X. Liao, and Z. Yi (Eds.): ISNN 2005, LNCS 3497, pp. 885–890, 2005. c Springer-Verlag Berlin Heidelberg 2005

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Existing Yield Curve Fitting Algorithms

To verify the merits of our proposed nonparametric and computational method for ﬁtting the yield curve, the performance of this method needs to be measured against widely used alternative models. In this paper, a smoothed spline model is used for this purpose. It assumes that the forward rate curve, δ, is a linear combination of basis functions. Cubic B-splines, a linear nonparametric interpolation method, are considered as basis. Let {Pi }1≤i≤N be the ith bond price at settlement date. δ is chosen to minimize the following objective function N

{Pi − Pˆi (δ)}2 + θ

τK

[δ (t)] dt 2

(1)

0

i=1

over the space of all cubic B-splines with knot points(time) τ0 < τ0 < · · · < τK . Pˆi (δ) is the ith theoretical bond price from the estimated yield curve [14]. Another approach to be considered is Nelson and Siegel curves [1], [2]. The forward rate curve modeled by Nelson and Siegel is δ(τ ) = β0 + (β1 + β2 τ )e−kτ

(2)

where β0 , β1 , β2 , and k are parameters to be estimated in the following way: N ∗ ∗ ∗ ∗ 2 Υ |Υ = (β0 , β1 , β2 , k ) = arg min [Pi − Pˆi (δ)] (β0 , β1 , β2 , k)

3 3.1

i=1

The Proposed Method Phase I: RBFN-Based Initial Curve-Fitting Phase

In the ﬁrst phase, we train a RBFN to estimate the spot rate curve with a given bond information on prices and maturities. A (generalized) radial basis function network (RBFN) involves searching for a suboptimal solution in a lower-dimensional space that approximates the interpolation solution where the approximated solution F ∗ (w) can be expressed as follows: F ∗ (w; τi , T, L) =

l

wj Φj (τi )

j=1

=

l j=1

wj φ(

τi − tj ) λj

(3)

where the set of RBF centers, T = {tj |j = 1, . . . , l}, and the set of scale parameters, L = {λj |j = 1, . . . , l}, are given [7]. Hence the training procedure of the GRBFNs is composed of selecting centers of the hidden neurons and estimating the weights that connect the hidden and the output layers. Once centers have

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887

been ﬁxed, the network weights will be directly estimated by using the least squares algorithm. To apply RBFNs to estimate the spot rate, we ﬁrst calculate the spot rate (ri ) with respect to maturity (τi ) on the basis of bootstrapping through bond information. Then an initial yield curve is ﬁtted as minimizing the criterion function N 1 ri − F ∗ (w; τi , T, L)2 . (4) J(w) = 2 i=1 The network weights can be directly estimated by using the pseudo-inverse, w = (ΦT Φ + λΦ0 )−1 ΦT R

(5)

where R is the vector of the spot rates obtained from the bootstrapping method, Φ = [φ(τi , tj )]i=1,...,N, j=1,...,K , Φ0 = [φ(τi , tj )]i,j=1,...,K and λ is a regularization parameter of the generalized RBFN [9]. The estimated spot rates, obtained from the trained RBFN, can be used to predict bond prices. Because the ultimate aim of the yield curve ﬁtting is to construct a bond pricing model, in the second phase, we have to additionally optimize the weights of the trained RBFN to minimize bond pricing error. 3.2

Phase II: Trust Region-Based Optimal Yield Curve Search

The second phase solves an unconstrained nonlinear programming to minimize the bond pricing error as follows: min E(w; R, T, L) = w

N 2 Pi (r(τ )) − Pˆi (F ∗ (τ ; w, T, L))

(6)

i=1

where Pi is the ith bond price for the true spot rate r(τ ) at time τ and Pˆi is the ith bond price for the spot rate F ∗ (w) obtained from the initially trained RBFN in the ﬁrst phase. To minimize Eq. (6) the second phase employs a trust region algorithm as follows. For a given weight vector w(n), the quadratic apˆ is deﬁned by the ﬁrst two terms of the Taylor approximation to proximation E E at w(n); 1 ˆ E(s) = E(w(n)) + g(n)T s + sT H(n)s 2

(7)

where g(n) is the local gradient vector and H(n) is the local Hessian matrix. A trial step s(n) is then computed by minimizing (or approximately minimizing) the trust region subproblem stated by ˆ subject to min E(s) s

s2 ≤ ∆n

(8)

where ∆n > 0 is a trust-region parameter. According to the agreement between predicted and actual reduction in the function E as measured by the ratio ρn =

E(w(n)) − E(w(n) + s(n)) , ˆ ˆ E(0) − E(s(n))

(9)

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Gyusik Han, Daewon Lee, and Jaewook Lee

∆n is adjusted between iterations as follows: s(n)2 /4 if ρn < 0.25 if ρn > 0.75 and ∆n = s(n)2 ∆n+1 = 2∆n ∆n otherwise

(10)

The decision to accept the step is then given by

w(n + 1) =

w(n) + s(n) if ρn ≥ 0 w(n) otherwise

(11)

which means that the current weight vector is updated to be w(n) + s(n) if E(w(n) + s(n)) < E(w(n)); Otherwise, it remains unchanged and the trust region parameter ∆n is shrunk and the trial step computation is repeated [8], [10], [11]. The proposed method has several advantages. At ﬁrst, since RBFN has a universal property of approximating an arbitrary curve, it can provide a good ﬁtting for the true yield curve. In addition, the use of a trust region-based search procedure makes much better estimation for the curve ﬁtted by RBFN.

4

Experimental Results

In this section, we verify how well the proposed method works compared with three existing ﬁtting algorithms of the yield curve which include – Cubic B-splines (Fisher’s method; FM): [14] – Nelson-Siegel curve (NS): [1], [2] – MLP method (MM) : [4], [6], [8], [9] The data used in the empirical analysis are 57 U.S. Treasury securities settled on November 3, 1997. U.S Treasury securities are semiannual-coupon bonds issued in order for U.S. Treasury Department to ﬁnance government borrowing needs at the lowest cost over time. The 57 securities have two quoted prices, bid price quotes and asked price quotes. We regard bid price quotes as real clean prices of the securities and use them in the empirical analysis. We used the spot rates bootstrapped from 50 of the 57 securities for training models, and the remaining 7 bonds for testing them. The kernel function for the proposed method is the Gaussian kernel function. The MLP method employed a multilayer-perceptron network instead of RBFN in Phase I. Simulation results are shown in Table 1 and Fig 1. The ﬁtting performance measures are the training error and the test error for bond pricing. Not surprisingly, NS, a parametric approach, provides the poorest performance because error is relatively severe. In eﬀect, NS is used when good accuracy is not required. The proposed method shows a better performance compared to the others in terms of accuracy.

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0.068 NS curve

Bootstrapped spot rate

0.066

FM’s curve

0.064

yield(r)

0.062 0.06 Proposed 0.058 0.056 0.054

MM’s curve

0.052 0.05

0

5

10

15 Time(τ): year

20

25

30

Fig. 1. Comparison of results for the ﬁtting of the yield curve Table 1. Simulation Result using four yield-curve ﬁtting techniques. (b.p.(basis point)=0.01%) Algorithm Training Error Test Error NS 1438 b.p. 97 b.p. FM 253 b.p. 73 b.p. MM 1219 b.p. 80 b.p. Proposed 287 b.p. 56 b.p.

5

Conclusion

In this paper, we’ve proposed a novel yield curve ﬁtting algorithm to mitigate pricing errors and improve a generalization performance for estimating the yield curve. The proposed method consists of a RBFN-based initial curve-ﬁtting phase and a trust region-based optimal yield curve search phase. In order to show the eﬀectiveness of the proposed algorithm, we’ve conducted experiments with 57 U.S. Treasury securities. The experimental results demonstrate that the new algorithm not only successfully improves generalization but also has better pricing accuracy compared with the existing algorithms for the ﬁtting of the yield curve. An application of the methods to other assets’ yield curve estimation remains to be further investigated.

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Acknowledgement This work was supported by the Korea Research Foundation under grant number KRF-2004-041-D00785.

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