JSTOR

I.  Introduction

The term structure of interest rates provides a characterization of interest rates as a function of maturity; it is defined as the pricing relationship that exists at any point in time between default‐free pure discount securities arranged by maturity. Term structure has a very important role in economics, serving at the macroeconomic level as a key transmission link between the monetary and real sectors and, at the microeconomic level, as an instrument of valuation of many financial claims.

The term structure of interest rates is formally defined as follows: assume that at time there exist zero coupon bonds with a full spectrum of maturities . Let their price and their yield be given by . Then the spectrum of yields is called the term structure of interest rates.

Alternatively, the discount rates can be defined representing the present value of one monetary unit repayable in years. So using discount rates, the term structure is the present value function , that is, the discount applied to a unitary payment to be made t periods hence.

If there is a liquid zero coupon bond market we can plot the yields from these bonds if we wish to construct the term structure of interest rates. However, coupon bond prices could be fit by a linear regression that adds up the values of the payments, just as easily as fitting zero coupon bond prices directly.

This work is focused on improving the polynomial spline estimation of the zero‐coupon term structure employing polynomial splines. Uses of polynomial spline functions to fit data prices have a long tradition in modeling interest rate term structure. In general, it is accepted that with a suitable choice of the number and the location of the knots, and employing low order polynomials, splines achieve great resolution along the whole maturity period. The seminal work in this direction was McCulloch (1971), who introduced the methodology of fitting the discount function by polynomial splines as a continuous function of maturity time. For cubic or higher‐order splines, the forward rates are smooth functions. Since the model is linear in the discount function, it is possible to use ordinary least squares (OLS) regression techniques. McCulloch suggested choosing a number of knots approximately equal to the square root of the number of bonds and placing the knots in such a way that each subinterval between two knots contains approximately the same number of observed maturities.

There are several extensions of McCulloch’s spline methodology. Langetieg and Smoot (1981) include fitting cubic splines to the spot rates rather than the discount function and varying the location of the spline knots. Complex nonlinear estimation procedures are required to implement these models. Shea (1984) draws attention to the pitfalls in smoothing the interest rate term structure using McCulloch’s spline functions. Splines cannot yield reasonable estimates without the intelligent use of constraints. The user must choose the number and location of polynomial pieces that will serve as the building blocks of the spline model. The polynomial order and degree of continuity of the spline function is a matter of choice. Finally, spline models may have to be constrained to achieve identification.

An important shortcoming in estimating the zero‐coupon term structure using polynomial splines is that estimations obtained by spline methods are sensitive to the choice of the numbers and location of the break points (knots) separating the splines. So, the knot selection can seriously affect the fit of the term structure of interest rates.

In this sense, an alternative literature has emerged proposing the use of parsimonious models. This new class of models supposes a unique functional form for the discount function on the whole range of maturities. This methodology avoids the problem of knot selection and imposes monotonicity a priori on the discount function.

Looking for monotonicity and parsimony, Vasicek and Fong (1982) present a different approach, namely, exponential spline fitting, exhibiting both sufficient flexibility to fit a wide variety of shapes of the term structure and sufficient robustness to produce stable forward rate curves. Vasicek and Fong claim that the exponential spline functions are superior to polynomial spline models. Nevertheless, Shea (1985) suggested that exponential splines yield forward rates that are unstable and fluctuate much like forward rates obtained using polynomial splines and recommended that ordinary spline techniques be used in preference to exponential splines.

In this article, a different approach is considered for selecting the spline basis using free‐knot splines. This approach provides a new solution to the historically difficult problem in numerical calculus of finding the optimal spline knot locations. This new methodology uses heuristic optimization techniques based on genetic algorithms (GA). The GA approach to spline estimation has been initiated in the statistic literature in papers such as Pittman and Murthy (2000). Pittman (2002) presents a model that uses GA to determine the appropriate B‐spline model for a given univariate data set.

The main contribution of the present study concerns applying GA to term structure estimation. In this sense, a new approach that is different and conceptually simpler than Pittman (2002) is suggested. It combines a GA method for solving the optimal knot locations with an OLS method for solving the remaining parameters and uses a truncated power functions basis, which has the advantage in that deleting the basis function is the same as deleting the knot.

II.  Spline Estimation of the Zero‐Coupon Term Structure

In order to get a flexible approximation of the discount function , McCulloch suggested approximating as a spline function. It means that must be a piecewise rth‐order polynomial with continuous derivatives, but its rth derivative is a step function.

Compressible expositions of spline methodology are available, for instance, in DeBoor (1978) or in Hastie, Tibshirani, and Friedman (2000). So to approximate as a spline function means that and the coefficients have to be estimated. Given that , we must have also for all j.

The functions are selected in two ways. On the one hand, we select some of them as , for , but also we select some as for , where

The function defined in (3) is known as truncated power function with the first derivatives in the being null. It permits a good adhesion between the pieces of rth‐order polynomials in the point .

So the spline specification of the discount function follows the form which is called spline of order r with interior knots , and in them the rth derivative changes discontinuously.

An analogous spline specification of the spot rates curve could be provided following the form For instance, cubic splines with two knots are generated as a linear combination of the functions and they generate piecewise three‐order polynomials with continuous second derivatives. This spline function may be expressed in a more familiar way as follows: It is easy to show that the definition (4) of the function is equivalent to imposing the function on the following three conditions (Eubank 1999):

1.	is a piecewise r‐order polynomial in every interval .
2.	has the first derivatives continuous.
3.	has the r derivative discontinuous in the knots .

Given the knots vector , the coefficients of expression (4) or (5) are estimated by OLS, and this is often described using the terms OLS‐splines or regression splines.

III.  The Problem of Free‐Knot Spline Selection

In order to approximate the discount function using OLS‐splines, it is necessary to select the degree of splines and the number and locations of knots for the estimator. Quadratic and cubic splines are typically used in practice.

Selection of the number of knots is a more delicate problem because the placement of a knot permits the estimator to adapt more to the data in that region. When only a few knots are used, the estimator looks like a polynomial and will not recognize complex structures in data. As the number of knots is increased, this produces increasingly flexible and potentially less erratic estimators.

Selecting the placement and number of knots for regression splines can be a combinatorially very complex task. The simplest approach to knot selection is a heuristic one through visual inspection of the data. In general, there is a set of ad hoc rules for locating knots.

The key idea to place knots through visual inspection is that more knots will be needed where the function seems to change more rapidly because the estimator will need greater flexibility in such regions. Eubank (1999) proposes a set of ad hoc rules for locating knots, as follows:

	For linear splines , place the knots at points where the data exhibit a change in slope.
	For quadratic splines , locate knots near local maxima, minima, or inflection points in the data.
	For cubic splines , arrange the knots so that they are close to inflection points in the data and so that not more than one extreme point (maximum or minimum) and one inflection occurs between any two knots.

The visual method of spline knot selection has computational advantages, but the estimation of the fitting curve (the term structure of interest rates, in particular) is sensitive to selection. For the specific problem of interest rate term structure, McCulloch suggested choosing a number of knots approximately equal to the square root of bonds maturities (so the number of interior knots will be the square root of bonds minus two), and place it in such a way that each subinterval between two knots contains approximately the same number of observed maturities. Ad hoc rules for locating knots have the inconvenience of subjectivity, and the fitted curve may be sensitive to the selection of the number and location of the knots.

Given and the order of splines, an objective and optimal selection method of the location of the knots is always desirable. A free‐knot spline is a piecewise polynomial function where the knots are considered as parameters to be estimated. The approximation to functions by splines has long been known to improve dramatically if the knots are free parameters (Rice 1969). The free‐knot approximation adds flexibility, thus improving the splines' power. Nevertheless, free‐knot splines have not been as popular as might be expected because of several serious disadvantages related to finding the optimal locations and number of the knots, as we will see shortly.

Finding the optimal locations of the knots is a difficult problem that is typically dismissed as numerically intractable. If it is desired to obtain a data‐driven choice for both the location and the number of knots in free‐knot spline selection, it is necessary to resolve the minimization problem where m is the number of available maturities and we optimize with respect to the arguments .

In order to simplify the problem, it is possible to fix the discrete arguments and and to resolve the simpler problem: minimizing the residual sum of squares simultaneously with respect to the basis coefficients and knot locations. This nonlinear problem has been tackled using nonlinear least squares methodology, such as the modified Gauss‐Newton algorithm (Gallant and Fuller 1973). Nevertheless, the knots enter the model nonlinearly, there are numerous local optima in the residual sum of squares surface, and it is not possible to determine the global optimum. In addition, many of the local optima correspond to knot sets with replicate knots that allow no smooth behavior of the predicted curve. The difficulties are multiplied if the number of knots is also estimated, so free‐knot splines are typically dismissed as computationally intractable because algorithms attempting to find optimal knots will tend to produce coincidental knots, which implies a reduction in the number of continuous derivatives for the estimator. This phenomenon was called “lethargy” property by Jupp (1975, 1978).

IV.  Free‐Knot Spline Selection with Genetic Algorithms

Genetic algorithms (GA) are a class of adaptive search and optimization techniques, based on principles of natural evolution developed by Holland (1975) and extended by Koza (1992). GA tries to overcome problems of traditional optimization algorithms. A GA starts with a population of randomly generated solution candidates and then applies the principle of fitness to produce better approximations to optimal solutions. Promising solutions, as represented by relatively better performing solutions, are selected and are bred together through a process inspired by Mendel’s natural genetics. The objective of this process is to generate successive population solutions that are better suited to the optimization problem than the solutions from which they were created.

GAs have been applied to a variety of problems in a diverse range of fields. They are used most effectively in situations where the space of possible solutions to an optimization problem is too large to be handled efficiently by standard procedures or when the solution is in some sense badly behaved, such as nondifferentiablility or possessing multiple local extrema. Nevertheless, applications of GA in econometrics are scarce.

The GA will permit a natural solution to the free‐knot spline selection problem. It is important to bear in mind that a notable difficulty of the Gauss‐Newton algorithm is starting with initial conditions close to the optimum. GAs have the advantage that it is always possible to start with an initial well‐behaved solution when we employ a high number of chromosomes.

In this study, the particular situation where the number of knots and the degree of the spline are fixed is considered. So, in the empirical application to estimate the interest rate term structure the popular quadratic and cubic splines will be used.

Let us consider the optimization problem (7), where it is assumed that the variables and are fixed. In (7) we have two kinds of variables in the objective function of the problem. So, the variables and intervene in a linear way, while the variables intervene in a nonlinear way. In order to resolve problem (7), an approach is proposed that combines ordinary least squares and GA. For every selection of the variables for the GA, and will be estimated through ordinary least squares.

The GA that is able to resolve the free‐knot spline problem is outlined below. It combines a GA method for solving the optimal knot locations with an OLS method for estimating the remaining parameters. This algorithm is called the “modified GA of sorting and deleting chromosomes” because sorting operations are essential to select the initial random population, and also in steps 6 and 7, where the ordered knot structure of some chromosomes can be destroyed through genetic operations. Step 9 is also essential in this algorithm because it avoids the coinciding knots problem by deleting those chromosomes that include some knots that are very close together.

This GA has the following sequential steps:

1.	Select the degree and the number of knots of the splines.
2.	Start with an initial random population of n k‐dimensional vectors, each representing a chromosome ; the are frequently called genes. Then sort the set of genes (knots) inside every chromosome so that an ordered knot spline configuration is obtained. This wide selection of starting values is a powerful tool and contrasts with classical optimization algorithms such as the Newton and quadratic hill‐climbing methods, which often break down when their starting values are not selected carefully.
3.	Estimate the parameters and in problem (7) by ordinary least squares.
4.	Form a ranking of ordered chromosomes from lowest to highest cost with respect to the cost function in (7).
5.	Select the best half chromosomes which are better adapted, and discard (delete) the other half of the population of high cost chromosomes.
6.	Select chromosomes in pairs from the good fitted population to produce two new offspring solutions through the crossover operation (also known as recombination) which is outlined in the figure below: So a couple of sets of knots are selected and bred together through this operation that produces an exchange of genetic information providing a new couple of sets of knots that are better suited, with respect to the cost function (7), than the original ones. In order to accelerate the convergence process, this linear combination procedure is performed for all parameters to the right or to the left of some crossover point.
7.	Alter a small percentage of the genes in the list of chromosomes by means of random mutations, so that GA explores promising regions in the parameter space.
8.	Sort the genes inside the chromosomes. This sorting operation is essential in the free‐knot splines selection problem because the operations in steps 6 and 7 can destroy the ordered knot structure of some chromosomes.
9.	Discard (delete) those chromosomes verifying the relations , for some previously selected and for some j between 1 and . The discarded chromosomes in the last step are replaced with new random chromosomes, so that the initial number n of chromosomes will be maintained. This replacement can reinforce the random search of optimum carried out by mutations. Moreover, the decimated chromosomes can be replaced using others previously discarded in step 5.
10.	Before replacement, the genes are sorted again inside the new random chromosomes.
11.	Return to step 3 and repeat this process sequentially until some convergence criterion is satisfied. The stopping criterion is usually satisfied if either the population converges to a unique solution or a maximum number of predetermined generations is reached.

V.  Empirical Results in Zero‐Coupon Euro Market Bonds

In order to illustrate the free‐knot spline with GA procedure estimating the curve of spot rates, the methodology was applied to Euro market interest rates. The sample consists of zero‐coupon Euro market bonds with 40 observed maturities from 1 day to 10 years. The exact 40 terms of maturities were:

	1 day, 1 week, 1, 2, 3, 6, and 9 months,
	1 year, 1 year and 3, 6, and 9 months,
	2 year, 2 year and 3, 6, and 9 months,
	3 year, 3 year and 3, 6, and 9 months,
	4 year, 4 year and 3, 6, and 9 months,
	5 year, 5 year and 3, 6, and 9 months,
	6 year, 6 year and 3, 6, and 9 months,
	7 year, 7 year and 3, 6, and 9 months,
	8 year, 8 year and 6 months,
	9 year, 9 year and 6 months,
	10 year.

This information covers a period of 541 days from 4/7/1997 to 5/28/1999 and was provided by Reuters.

During this period, the term structure of interest rates was estimated every day by the free‐knot spline procedure using a number of interior knots, varying from (overall polynomial interpolation) to . In order to render possible comparisons, we have also estimated the curve of spot rates and the discount curve using McCulloch’s methodology. McCulloch (1971) suggested choosing a number of knots (interior knots plus two boundary ones) approximately equal to the square root of bonds in the sample, and placing them in such a way that and equals the maturity of the longest bond. Each subinterval contains approximately an equal number of observed maturities. In our sample, McCulloch’s methodology suggests six knots (four interior knots plus two boundary ones), where “six” is the nearest integer to the square root of the number of observed maturities ( ). McCulloch’s interior knots are placed at maturities of 1 year, 3 years, 5 years, and 7 years, respectively. Second and third degree polynomial functions were chosen in order to implement the splines.

In order to illustrate the GA methodology, the curve of spot rates corresponding to 3/27/1998, which matches the 250th day in the sample, was estimated. As in most estimations of this work, the GA employed had 100 chromosomes, the rate of mutation was set at 5%, the minimum distance permitted between consecutive knots was (step 9 in the GA) and a simple convergence criterion of reaching a maximum number of 20 iterations (generations) was established.

Figures 1 and 2 show the estimates of spot rates for quadratic and cubic splines . In these graphs the maturities are placed on the x‐axis, the asterisks represent the available zero coupon bonds, the solid circles represent spot rates estimated by McCulloch’s spline methodology, and the solid line represents free‐knot splines estimated with GA. In both cases, estimation was performed with four interior knots ( ). As observed, free‐knot splines with the GA methodology match the data almost perfectly, producing a better adjustment than McCulloch’s.

Tables 1 and 2 show the estimated free knots for the quadratic and cubic splines, for 11 values of , representing the number of knots ( ) corresponding to the curve of spot rates in date 3/27/1998. Note that corresponds to a polynomial adjustment. At the bottom of tables 1 and 2, the residual sum of squares (RSS) for both the GA and McCulloch methodologies are presented. Observe that the RSS of the free knot with GA splines are always lower than the McCulloch splines. Nevertheless, in order to compare residuals with the same number of free parameters, additional evidence is necessary. So, the usual goodness of fit measure based on the mean of RSS, usually called MRSS, is used: where and m is the sample size.

Observe that when comparing fixed‐knot splines to free‐knot splines, it should be kept in mind that whereas a fixed‐knot spline of degree r has parameters (when the intercept is estimated, as in this yield curves), a free‐knot spline has free parameters. A free‐knot spline with knots should therefore be compared to a fixed‐knot spline with knots, and so forth. Even adjusting in this way the number of free parameters, the free‐knot spline ordinarily outperforms the comparable fixed‐knot splines in eight out of 10 cases, as it is possible to observe looking at the MRSS goodness of fit measure at the bottom of table 1.

Similar evidence in favor of GA could be directly provided through the RSS. So the last but one row in table 1 shows that the GA model with , and therefore free parameters, actually does slightly worse than the McCulloch model with and, therefore, with the same number of free parameters ( ). However, the difference is quite small ( for McCulloch versus 0.0048 for GA). For 12 free parameters ( for GA or for McCulloch), GA does slightly better. In all other cases, however ( , , ), GA does much better than McCulloch. So, using the same number of free parameters, GA often does much better than McCulloch and never does more than slightly worse.

Table 2 gives unambiguous evidence in favor of GA. In table 2, GA in fact does better than McCulloch, even when the number of free parameters is held constant as above. Only for does GA do slightly worse than McCulloch attending to the MRSS criterion.

In order to obtain a more extensive comparison, both procedures were applied to all of the dates in the sample, estimating the spot curve and the discount curve during the available 541 days. The results are presented in table 3 (for the spot curves) and table 4 (for the discount curves), which correspond to quadratic and cubic splines, respectively. The average of the residual sum of squares for the 541 days of the sample is lower for the free‐knot splines with the GA methodology, both for the spot and discount curves.

Nevertheless, likewise in tables 1–2, the RSS comparison should be between free‐knot splines with k knots and fixed knot splines with 2k knots. Even with this adjustment, GA always outperforms the McCulloch spline in table 3. In table 4, GA usually outperforms, though there are a couple of cases in which it is (only slightly) worse.

Estimation techniques of interest rates using free‐knot splines with GA could have their shortcomings. The most important of these is that GA is a random optimization technique and, if we repeat the optimization process several times, we do not always have the same knots exactly. Nevertheless, a numerical example with our market data might make the potential benefits of free‐knot splines clearer. In this example, it will be shown that if we reestimate the knots 100 times, although the estimated knots change each time, as much the variance of every knot as the residual sum of squares go to zero if the number of chromosomes of the GA increase. Table 5 and table 6 show the standard deviations of four locations of knots and RSS of estimation obtained as a result of 100 reestimates of the spot rate corresponding to 3/27/1998 for quadratic and cubic polynomials, respectively. The columns of tables 5 and 6 represent the standard deviations of the estimated knots for a different number of chromosomes, from 800 to 25. In all cases the maximum number of generations allowed in the convergence criterion of the GA is 50.

As it is possible to observe in tables 5 and 6, if we take more than 100 chromosomes in the GA, although a random optimization algorithm to obtain the knot location is employed, the algorithm becomes robust because the standard deviations of the knots and the RSS decrease to insignificant levels. As a final observation, it is necessary to point out that all calculations in this article were implemented using MATLAB.

VI.  Conclusions

The term structure of interest rates provides a characterization of interest rates as a function of maturity. The use of polynomial spline functions to fit prices has a long tradition in modeling interest rate term structure. Nevertheless, a reliable spline approximation may depend crucially upon the selection of knots, so the choice of the number and position of knots seriously affects the quality of the fit in the estimated term structure. As selecting the placement and number of knots for regression splines can be a combinatorially complex task, previous research in estimating the term structure with splines used ad hoc rules to choose the knot sets.

Finding the optimal location for the knots is well known to be difficult and has been frequently dismissed as numerically intractable. So free‐knot splines are usually considered a rather difficult nonlinear optimization problem.

The computational difficulties of optimizing knot locations have long being investigated. In this article, a different approach to the free‐knot splines problem is proposed, providing a solution to the historically difficult problem in numerical calculus of finding the optimal spline knot locations. This new methodology uses heuristic optimization techniques based on genetic algorithms.

GAs are a class of adaptive search and optimization techniques based on the principles of natural evolution developed by Holland (1975). GAs try to overcome problems of traditional optimization algorithms. A GA starts with a population of randomly generated solution candidates, which apply the principle of fitness to produce better approximations to an optimal solution. Promising solutions, as represented by relatively better performing solutions, are selected and bred together through a process based on Mendel’s natural genetics. The objective of this process is to generate successive population solutions that are better suited to the optimization problem than the solutions from which they were created.

With the purpose of resolving the free‐knot spline problem, a modified GA of sorting chromosomes was presented in this article. The results suggest that GAs are able to resolve the free‐knot spline problem by determining the positions of critical points.

In order to illustrate this procedure in practical situations, the methodology was applied to estimating the spot and discount curves using samples of zero‐coupon Euro market bonds. Attending to the residual sum of squares and other goodness of fit measures, the results suggest that free‐knot splines with the GA methodology match the data almost perfectly and produce a considerably better adjustment than McCulloch’s. This holds for both quadratic and cubic splines and for a number of interior knots between one and 10. Besides, even when the number of free parameters is held constant, in most cases GA clearly outperforms McCulloch’s model, although there are a couple of cases in which it is (only slightly) worse.

There is an extension of this research. In this article, the degree r of the splines and the number k of knots were considered to be constant in optimizing the loss function. The resulting optimization problem considering r and k as arguments of the lost function are difficult even for GA because r and k dramatically affect the size of the chromosomes. However, this optimization problem can be dealt with using more flexible and sophisticated developments of GA called genetic programming (Koza 1992). Nevertheless, the parsimony principle points out that genetic programming would yield approximately the same results as the present GA approach without an extensive search of the degree of the splines.