Estimating Yield Curves of the U.S. Treasury Securities: An Interpolation Approach

Transcription

1 Estimating Yield Curves of the U.S. Treasury Securities: An Interpolation Approach Feng Guo a, a Chinese Academy of Finance and Development, Central University of Finance and Economics, China. Abstract Following the approach of interpolation, this paper proposes the multiple exponential decay model to fit yield curves based on both the U.S. Treasury inflation-indexed securities and nominal Treasury securities. Several estimation methods, including unconstrained/constrained minimization, quadratic programming, and iterative weighted least squares, are applied to estimate the unknown parameters according to different curve-fitting purposes. As a comparison, the paper runs a horse-race between the proposed model and the alternatives. The results show that the multiple exponential decay model successfully (1) outperforms competing models in terms of fitting errors, (2) realizes a parsimonious functional form, and (3) smooths through idiosyncratic variations associated with the forward rate curve. In addition, selection of the number of terms/parameters in a nonparametric interpolation model governs its overall effectiveness of fit, which is first optimized by three statistical tests in this paper. JEL classification: G12; E43; C52 Keywords: Term structure of interest rates, Yield curve, Interpolation, Treasury securities, Model evaluation Corresponding author. Tel: Address: 39 South College Road, Beijing , China. address: guo.117@osu.edu (Feng Guo) Preprint submitted to The 25th FINANCE FORUM March 6, 2017

2 1. Introduction A continuous yield curve, spanning from zero to the longest term, is of great help to interpret macroeconomic fluctuations, monetary policies, and speculation/hedging behavior. Estimating the yield curves, either in real or nominal term, is therefore a good place to start relevant research on those topics. Yield curve here refers to the discount function, the zero-coupon yield curve, or the forward rate curve since each is a transformation of the others. Basically, they are all derived from the benchmark yields of Treasury securities with different maturities. If the Treasury issued bills, notes, and bonds with a continuous spectrum of maturities, then we could simply observe yield curves implied by the full set of spot and forward rates. In practice, however, the U.S. Treasury has instead issued a limited number of securities with different maturities and coupons. Each of these can be viewed as a basket of zero-coupon securities: one for coupon payment at each semiannual coupon date and one for the principal payment at maturity. In general, the market does not have securities at all maturities and hence cannot simply solve for the implied yields. Instead, we must infer the missing bond yields across the maturity spectrum so as to retrieve a continuous yield curve. Basically, two lines of research have been trying to recover the missing bond yields. One is featured by applying interpolation or curve-fitting methods, the basic idea to which is that the price of any security is assumed to be governed by a smoothly fitted discount function, δ(m). Fama and Bliss (1987), McCulloch (1975); McCulloch and Kwon (1993); McCulloch and Kochin (2000), Nelson and Siegel (1987), and Fisher et al. (1995) follow this line, though these studies differ considerably in respective interpolation models and estimation techniques. The other approach in turn focuses on fully specified, dynamic term structure models. In that line of research, a prevailing approach is to build up reducedform term structure models in which yields are expressed as constant-plus-linear function of some state variable(s); both of the state variables and coefficients follow dynamic stochastic processes. This kind of term structure models is well-known as Affine Term Structure Model (ATSM), evolved through the one-factor models such as Vasicek (1977) and Cox et al. (1985), and established/refined by Duffie and Kan (1996), Dai and Singleton (2000), Piazzesi (2003), etc. A successful element of ATSMs is the cross-asset restrictions imposed on the model to 2

3 eliminate arbitrage opportunities. Nevertheless, a number of desirable properties make the static approach, curve-fitting methods, irreplaceable in studying term structure of interest rates. First, interpolation models are independent of any equilibrium settings. This type of models is pure descriptive in that the yield is a function of term to maturity only. Second, this approach allows simple, 35 parsimonious functional forms to represent a wide range of shapes generally associated with yield curves. Nelson and Siegel (1987) advocate parsimonious yield curve models succeeding in the objective that the whole term structure of yields can be described more compactly by a few parameters. Third, the simple functional forms, supplemented by improved estimation techniques, reduce computational burden effectively and hence can realize high-frequency 40 fitting/re-fitting. This single advantage makes curve-fitting methods irreplaceable in financial industrial applications. An interpolation typically begins with specifying a functional form either to approximate discount function or forward rates, and then estimates the unknown parameters. An extensive body of literature hence has contributed to develop favorable functional forms as 45 well as efficient estimation techniques 1. Among them, two popular approaches exhibit great potential to facilitate high frequency fit with a full spectrum of maturities: one is McCulloch and Kochin (2000); the other is the extended Nelson-Siegel (Nelson and Siegel, 1987; Svensson, 1994). The former proposes a Quadratic-Natural (QN, henceforth) cubic spline functional form to fit a negative log discount function, whose parameters are estimated by 50 a well-designed iterative procedure based upon linear least squares. The latter instead fits the forward rate curve via a set of exponential decay functions and nonlinear minimization methods. Gürkaynak et al. (2007) point out that neither of the two approaches wins over the other; the selection of approximate functions largely depends on the purpose of yield curve fitting. They argue that two functional forms differ in flexibility each allows the fitted 55 yield curves have. The cubic spline approach, according to their argument, brings more flexibility on the shape of a yield curve and is thus good for financial practitioners who are looking for small pricing anomalies. In contrast, macroeconomists may prefer the more 1 Bliss (1996) gives an in-depth survey of the literature. 3

4 parsimonious exponential decay model because a relatively rigid forward curve that smooths through idiosyncratic variations helps investigate the fundamental determinants of the term structure of interest rates. Unfortunately, there is no further research trying to integrate the advantages of two approaches and put up with new functional forms. This becomes a major motivation of this paper. Specifically, I am trying to answer three questions in constructing a new interpolation model for yield curve fitting. First, what target do we choose to approximate in order to ensure an asymptotical convergence of the fitted curves? Second, what kind of functional forms should the interpolation model be based on after considering both economical meaningfulness and the practicability in real applications? Third, how do we estimate the parameters in the proposed model to obtain the best efficiency? For the first question, McCulloch (1975), and McCulloch and Kwon (1993) approximate the discount function directly. By modeling a discount function, they derive a linear pricing function for coupon-bearing securities, which makes the estimation largely easier. However the behavior of the fitted yield curve at long-end will become hard to govern. This problem is solved by another approach: instead of targeting on discount functions, the bond pricing function can be specified in terms of the logarithm discount function with well designed end-point conditions. Nevertheless, modeling a log discount function results in a nonlinear pricing function, whose estimation becomes relatively complicated. McCulloch and Kochin (2000) and Nelson and Siegel (1987) actually follow this approach. The second dimension is to specify an interpolation functional form for the target function selected. Again, two approaches are competing in this area. The QN cubic spline approach, introduced by McCulloch and Kochin (2000), features a nonparametric model. It allows substantial flexibility on the shape of yield curves, but we do not need too much on most time. Nelson and Siegel (1987), Svensson (1994), and Gürkaynak et al. (2007, 2010) in turn specify an exponential decay functional form. In contrast to the cubic spline, the exponential decay form results in a parametric and parsimonious model, which, though depicting a relatively rigid forward rate curve, is able to smooth through idiosyncratic variations generated by individual securities. The two approaches also differ in their long-end behavior. If the log discount function is targeted, cubic spline functional form needs an extra end-point condition; 4

5 otherwise, the yield curve or the forward rate curve will not necessarily show asymptotical convergence. However, no extra condition is required for the exponential decay function 90 because of its natural property of asymptotic decay. Estimating the unknown parameters is a less popular issue in the literature; however, various approaches do differ substantially in terms of computational efficiency and effectiveness. In addition, a single estimation method is unable to meet all curve-fitting purposes. Unfortunately, previous literature fails to answer this question or even consider this issue 95 exclusively. In order to answer the three questions, I first propose a multiple exponential decay functional form to fit the negative log discount function. Then, several estimation methods are introduced to estimate unknown parameters in an objective function according to two distinct curve-fitting purposes. As a comparison, I extend McCulloch cubic spline model 100 to fit the same data set. Since McCulloch and Kochin (2000) pin down the number of terms/parameters in their QN cubic spline model arbitrarily, I establish a procedure to optimally adjust the nonparametric model specification to specific data sets. The remainder of this paper is generally methodological and hence organized as follows. Section 2 gives the fundamental definitions of yields and briefly reviews basic bond maths. 105 Section 3 proposes the interpolation model based upon exponential decay basis functions. I also compare it with two competing model specifications in the same line of research. Section 4 discusses the estimation methods for unknown parameters in the interpolation model and compares the results obtained through different methods. Section 5 extends the estimation methods in Section 4 for financial practitioners who in turn look for investment 110 opportunities. Section 6 concludes. 2. Identities and Definitions The concepts of yields and bond pricing are basic for readers in the field of fixed income. The purpose of this section is however to establish common notations and definitions for the remainder sections. 115 Any pricing problem for (option-free) fixed income assets is based on a discount function. Let δ(m) denote the discount function as a function to maturity m. It gives the present value 5

6 120 of $1 to be paid in m years ahead, and immediately follows that δ(0) = 1, and δ( ) 0. Given the discount function one can price any fixed income securities with distinct styles of cash flows. For example, the value of a hypothetical bond with continuous coupon payments, a fixed annual coupon rate c, years to maturity m, and the principal payment $1 at maturity, is calculated as the sum of the present value of its individual payments: p = δ(m) + c m 0 δ(s) ds. (1) 125 One can also derive the spot return of this bond market, or the yield, once the discount function is known. Among the conventional concepts of yields, the zero-coupon yield/spot rate is the most fundamental and mathematically simplest concept. The continuously compounded zero-coupon yield is defined as: log δ(m) y(m) = m. (2) 130 Except for Treasury bills (T-Bills, henceforth) which are sold at discount with a maturity less than one year, all other Treasury securities, including Treasury notes, Treasury bonds, and Treasury Inflation-Protected securities (T-Notes, T-Bonds, and TIPS respectively, henceforth), are coupon-bearing securities. A prevailing way to quote coupon-bearing yields is through par bond yields, which are the coupon rates at which a security with a maturity m would sell just at par (and hence have a coupon-equivalent yield equal to that coupon rate). The continuously compounded par bond yield with maturity m is given by: m 0 y p (m) = 1 δ(m) (3) δ(s) ds. A term structure of interest rates can also be expressed in terms of forward rates. A forward rate is the forward yield on a security, at which an investor would agree to make an investment today over a specified period in the future for m 2 years beginning m 1 years hence. The continuously compounded return on this forward contract can be expressed as follows: f(m 1, m 2 ) = 1 ( ) δ(m1 + m 2 ) log m 2 δ(m 1 ) = 1 m 2 [(m 1 + m 2 )y(m 1 + m 2 ) m 1 y(m 1 )] 6

7 As m 2 approaches to zero, we obtain the instantaneous forward rate, implying the instantaneous return m years from today that an investor would demand today. Mathematically, 135 which follows f(m) f(m, 0) = y(m) + my (m) = ( log δ(m)) = δ (m) δ(m), (4) y(m) = 1 m m 0 f(s) ds, and f(m 1, m 2 ) = 1 m 2 m1 +m 2 m 1 f(s) ds. (5) Eq. 1 through 5 show that each of the spot rate, the par yield, or the forward rate is a transformation of the others; they are all based on the discount function δ(m). In this sense, the curve-fitting problem becomes a problem to specify and estimate a discount function Model In this section, I propose a brand new curve-fitting model and compare it with alternative interpolation models in the same line of research. The properties of each model as well as their distinctions will be discussed. The estimation and results are in next section Discount Function v.s. Log Discount Function 145 According to Section 2, the yield curve fitting is to find an explicit discount function. However, direct modeling on a discount function is not a preferred approach 2 because it is hard to regulate the behavior of that discount function, particularly at its long-end. Theoretically, a discount function curve should decline monotonically and decay to zero asymptotically. Yet an empirical discount function, with parameters estimated by actual market 150 data, is likely to violate its theoretical definition by entering negative region as m becomes large. In order to avoid this problem, some researchers choose to target on a (negative) log discount function instead. This alternative approach is able to govern the model s behavior across maturities by imposing appropriate end-point conditions. For example, McCulloch and Kochin (2000) use a long-end natural condition in a cubic spline model to let the log 155 discount function monotonically increase beyond the longest maturity outstanding. 2 Since McCulloch and Kwon (1993), there has been no research following this approach. 7

8 160 Of course, there are advantages to construct a discount function. Particularly, if the target function is a discount function, the corresponding bond pricing function becomes linear in unknown parameters, whose estimation is relatively easier. In contrast, a log discount function model results in a nonlinear bond price function, which causes computational complexity. Nevertheless carefully designed estimation methods, such as Iterative Linear Least Squares in Section 4.2, will largely reduce the degree of complexity Model Specification 165 Considering the pros and cons of two target functions, this paper chooses to target on a (negative) log discount function. The general functional form φ(m) is given by: k φ(m i ) log δ(m i ) = β j Ψ j (m i ) i = 1,..., n j=1 s.t. δ(0) = 1 or φ(0) = 0, Ψ j (0) = 0; (6) k n, which is a linear combination of Ψ(m) basis functions with parameters β, and subject to the conditions that the fitted φ(m) goes through the origin and the total number of its parameters k is less than n, the total number of effective securities. The effective securities are those outstanding securities on a market selected to be used in curve-fitting. Refer to AppendixA for the selection process. Apparently, the model is nonparametric since k is not fixed. Two heuristic facts help to specify the form of the general function φ(m) and the basis function Ψ j (m). First, the discount function δ(m) features approximately an exponential decay curve: some shapes (monotonic decline, humped, or S shaped) at short-end and an asymptotical decay at long. The negative log discount function φ(m) in turn should be (approximately) monotonically increasing. The linear functional form in Eq. 6 thus keeps in line with this feature. Second, according to ATSM, yields y(m) are generated by a differential equation; the instantaneous forward rate f(m) thus takes an exponential functional form because it turns out to be the solution to y(m) by Eq. 5. According to Eq. 4, f(m) is the first-order derivative of φ(m), which should also take exponential forms. The basis function Ψ j (m) therefore ought to be specified by exponential functions. 8

9 In addition, there are several properties we expect Ψ j (m) to have. In order to govern the long-end behavior of various yield curves, the first-order derivative of one Ψ j (m) should be one, and all the other Ψ j (m) die out to zero at infinity; their first-order derivatives also die out to zero so that the forward rate curve would have an asymptote and the zero-coupon yield curve have another asymptote. Given all these features, I specify Ψ(m) as: 185 ) 1 exp ( mτj if j = 1,..., k 1 Ψ j (m) = (7) m if j = k where parameter τ j is constant to maturity m but specific to Ψ j (m). The functional form of Ψ j (m) in Eq. 7 is associated with a concept called E-fold Life, which is defined as the timescale ) for a quantity to attenuate to 1/e of its previous value. As m increases from below, exp ( mτj will decline to 1/e as m = τ j. It thus follows that different values of τ j lead to different rates of decay. A small τ j realizes rapid decay in the regressors 190 and therefore will be able to fit a curvature at short maturities well. In contrast, a larger τ j produces slower decay in the regressors that can only fit curvatures over longer maturities. On one extreme, Ψ j (m) converges at once as τ j = 0. As τ j, on the other extreme, Ψ j (m) takes forever to converge. When τ j [0, ), the rate of convergence monotonically declines in the value of τ j, and the function generates distinct curvatures. Since k determines 195 the number of curvatures the function is able to feature, by properly introducing k terms of Ψ j (m) into φ(m), the model can fit any number of shapes for the yield curves at exact maturities needed Model Comparison I: An Qualitative Analysis A comparison between the multiple exponential decay model and two relevant interpola- 200 tion models is conducted: one is the extended QN cubic spline interpolation; the other is the extended Nelson-Siegel. All the models share the same φ(m), whereas their specifications of Ψ(m) are different. McCulloch and Kochin (2000) fit a QN cubic spline to a negative log discount function. They select a small number of well-spaced maturities (as knots) from a large pool of T- 205 Notes & -Bonds to fit a just-identified yield curve. Specifically, ν + 1 knots {κ j } ν j=0 are 9

10 210 selected, where κ 0 0 and κ ν m n. In order to let the spline model be extrapolated linearly beyond the longest maturity outstanding, they impose a natural restriction on the last knot: φ(m) = 0, for m [m n, ). At the short-end, they impose a quadratic restriction instead, rather than a second natural restriction, to avoid the counterfactual outcomes: y (0) = 0 and f (0) = 0. The quadratic restriction on the first interval is given by: φ(m) = 0, for m [0, m 1 ]. They thus name their spline model Quadratic-Natural cubic spline. This model is specified as: Ψ j (m) = θ j (m) θ j (m n ) θ ν+1(m n ) θ ν+1(m), j = 1,..., ν (8) where θ 1 (m) = m θ 2 (m) = m 2 θ j (m) = max{0, (m κ j 2 ) 3 }, j = 3,..., ν There are several disadvantages associated with this model specification. First, although the spline-based model is flexible enough to fit various shapes for yield curves, the cubic function tends to be too sensitive to idiosyncratic fluctuations. Idiosyncratic issues arise for microstructure reasons, such as auction cycles, liquidity premia, hedging/speculation demand, or repo market specialty. According to Gürkaynak et al. (2007), the spline-based model enables too much flexibility to the fitted curves and hence turns out to be not very helpful in understanding the fundamental determinants of the yield curve. Second, the two artificially imposed restrictions add extra unwanted features to the model. The purpose of imposing end-point conditions is to pin down all unknown parameters in an exact-fit interpolation model. Unfortunately, the natural restriction results in a linearized forward rate curve at long-end, which in turn leads to hypothetical arbitrage opportunities 3. The short-end quadratic restriction is actually an expedient in the absence of a second natural restriction. It arbitrarily assumes that the log discount function takes quadratic form within its first interval, which has no theoretical foundation. Admittedly, certain limitations inevitably bring on the shortcomings aforementioned. 3 Since there is no Treasury security with longer maturity beyond the longest outstanding, the hypothetically existed arbitrage opportunities are not realizable. 10

11 Back to one and a half decades ago when the method was first proposed, only a handful of securities with distinct maturities in TIPS market were outstanding. This constraint made 230 fitting an over-identified real yield curve impossible. McCulloch and Kochin (2000) therefore had to impose a quadratic restriction for an exact fit. In order to maintain consistency, they choose to use the same set of maturities in TIPS market for the estimation of the nominal yield curves from over 150 U.S. T-Bonds, T-Notes, and T-Bills outstanding. Later, even though more TIPS securities were issued, the authors did not adjust the model so that all 235 fitted curves were historically comparable. This gives rise to the problem of biased estimation: with a fixed number of knots, the more securities outstanding, the more biased the estimated term structure could be in that only those securities selected as knots 4 enter in parameters estimation, while the price information in other securities are simply ignored. Accordingly, I extend the QN cubic spline model to partially revise McCulloch and 240 Kochin (2000). First, I select k + 1 maturities as knots {κ j } k+1 j=1 by assigning the first knot to be zero, the last to be the longest maturity outstanding, and the remaining k 1 knots evenly distributed in the maturity spectrum 5. Then, I impose a natural restriction on the long-end in the same way as that of McCulloch and Kochin (2000), which becomes the only restriction in the extended model. Since the quadratic restriction at short-end is eliminated, 245 I call this extended model Semi-Natural cubic spline. The basis function Ψ(m) is hence specified as: Ψ j (m) = θ j (m) θ j (m n ) θ k+2 (m n) θ k+2(m), j = 1,..., k + 1 (9) 4 Instead of selecting on-the-run Treasury securities only, the securities/knots selection in McCulloch and Kochin (2000) is rather arbitrary. 5 Mathematically, if we observe n maturities { in a bond market ranked in an ascending order from the shortest {m i } n i=1, then κ 1 = 0, κ k+1 = m n, and κ j = m 1 + (j 1) (m } n m 1 ) j = 2,, k. Except for k the two end-point knots, the knots may not overlap outstanding maturities, but be definitely within the overall maturity spectrum. 11

12 where θ 1 (m) = m θ 2 (m) = m 2 θ 3 (m) = m 3 θ j (m) = max{0, (m κ j 3 ) 3 }, j = 4,..., k Apparently, the hypothetical arbitrage opportunities are still present because we have to impose a long-end natural restriction at least in order not to let the cubic function diverge to infinity at large. In a vivid contrast, the multiple exponential decay function does not need any artificial restrictions due to the exponential decay inherent in it. The second relevant model for comparison is the extended Nelson-Siegel. The model was first proposed by Nelson and Siegel (1987), and further developed by Svensson (1994). This class of models originally targets on forward rates 6. Gürkaynak et al. (2007, 2010) summarize the model as: f(m) = β 1 + β 2 exp ) ( ) ) ( ) ) ( mτ1 + β 3 exp ( mτ1 mτ1 + β 4 exp ( mτ2 mτ2. (10) In order to be comparable, I integrate f(m) according to Eq. 4 and obtain another expression in terms of a negative log discount function, φ(m). The general functional form is identical to Eq. 6, while the specification of the basis function Ψ(m) is given by: Ψ 1 (m) = m, ) Ψ 2 (m) = m exp ( mτ1, ) Ψ 3 (m) = 1 exp ( mτ1, ) ( )) Ψ 4 (m) = m exp ( mτ2 + τ 2 1 exp ( mτ2. (11) 260 Several features associated with this model are apparently observed. Firstly, it allows forward rates begin at horizon zero at the level β 1 + β 2 and eventually asymptote to β 1. Second, it is a parsimoniously parametric form, in which a fixed number of β and τ are present. In addition, two humps are allowed in the forward curve, the locations of which are determined by τ 1 and τ 2. Gürkaynak et al. (2007, 2010) argue that a yield curve often 6 The paper refers instantaneous forward rate to forward rate henceforth for convenience. 12

13 needs two humps, one at short maturities associated with monetary policy expectations and the other at long to capture convexity effects. It is hence apparent that Ψ 1 (m) in Eq determines a long-term horizon level while Ψ 2 (m) to Ψ 4 (m) characterize the short- and medium-term shapes: monotonic, curvatures, or humps. The extended Nelson-Siegel and the multiple exponential decay model share some common elements in modeling the Ψ(m); however, they differ considerably in the flexibility each model allows the fitted yield curves to have. For the former, the coefficient k is fixed at 4, 270 i.e. only four Ψ j (m) enter the interpolation function with six unknown parameters four βs as well as two τs. This results in a rather rigid yield curve for some data sets. As aforementioned, it must assign a fitted yield curve two humps, which could be redundant for a real yield curve and scarce for a nominal one. The multiple exponential decay function in turn does not have this restriction since it lets any k terms of Ψ(m) enter so as to introduce equal 275 or even richer shapes to the fitted yield curves 7. This feature greatly enhances the fitting flexibility of the model. In sum, the multiple exponential decay function has qualitative advantages over alternative models in the same line of research. Compared with cubic spline models, it is more parsimonious, more robust to idiosyncratic variation, and free from arbitrage opportunities 280 caused by end-point restrictions. Compared with the extended Nelson-Siegel, the new model offers more flexibility and less fitting errors to the fitted yield curves. These qualitative advantages will be further demonstrated in next section with empirical results Model Comparison II: An Quantitative Analysis Quantitatively, the multiple exponential decay model enables better curve-fitting than 285 the alternatives in terms of fitting errors. Following Ioannides (2003) and Jordan and Mansi (2003), I compute the monthly weighted root mean square errors (RMSE) between the estimated and the actual bond prices with the weight being the inverse of the modified duration. In accordance with the literature, the monthly price data is given by the market quotations on the last business day of each month. Table 1 reports RMSE statistics generated by three Although each Ψ j (m) in Eq. 4 is just able to generate a single curvature, by controlling over the parameters τ j in a combination of Ψ j (m) it is able to generate a hump or even more complex shapes. 13

14 interpolation models respectively on monthly data for both the U.S. real (based on TIPS) and nominal (based on T-Bills, -Notes, and -Bonds) markets over a period between April and December 2014, a total of 201 months. It shows that the multiple exponential decay model not only generates the smallest cumulative fitting error over the data period, but also has the lowest volatility. The maximum value of individual monthly price errors generated by the new model does not exceed that of other models either. The cubic spline model and the extended Nelson-Siegel on the other hand perform relatively weakly in terms of the cumulative sum of RMSE; the latter slightly outperforms the former in fitting a real yield curve, but underperforms in the case of a nominal yield curve. The worst performance demonstrated by the extended Nelson-Siegel in fitting the nominal curves may suggest that the nominal U.S. Treasury security market with approximately 5 to 10 times more outstanding securities than the U.S. TIPS market naturally requires a nonparametric interpolation model with a variable (instead of a constant) k. [Table 1 about here.] Figure 1 outlines the evolution of the price-based monthly weighted RMSEs over the entire data period for both markets. For the real Treasury market illustrated by Figure 1-(a), the multiple exponential decay model outperforms the alternative models for almost every single month over the entire time series. The semi-natural cubic spline performs better than the extended Nelson-Siegel prior to 2008, but does worse after that. The three models however seem to have a similarly upward trend over time in the real-term market, which should be explained by the increasing number of outstanding TIPS securities from 4 in April 1998 to 38 in December This kind of time series pattern is not seen in the nominal market, illustrated by Figure 1-(b), where the number of nominal securities outstanding has been relatively stable within the recent two decades. What can be seen in Figure 1-(b) is the strikingly weak performance given by the extended Nelson-Siegel. [Figure 1 about here.] 8 The data period starts in April 1998 because the first 30-year long-term TIPS was not issued and traded until then. 14

15 In addition, the goodness of fit can be further scrutinized in different maturity ranges. Table 2 reports that the multiple exponential decay model outperforms the competing models in all individual maturity ranges from 0 to 30 years at both real and nominal markets 9 by having the smallest cumulative fitting errors. This evidence further justifies the quantitative 320 advantages of the multiple exponential decay model over its competitors. [Table 2 about here.] The estimated bond prices are key to the quantitative comparison above. Particularly, all three models are estimated by Iterative Linear Least Squares (ILLS), which is to be introduced in Section 4.2 and AppendixB. The estimation of the nonparametric models 325 involves picking an optimal value for the parameter k, which is based on a series of tests in Section Estimation and Results According to the model specification in Eq. 6 and 7, three groups of parameters, β, τ, and k, are to be estimated. In this section, I introduce the estimation methods for each of 330 the three, and show the estimated yield curves. In estimating τ and β, I assume k is given. Then I will discuss the technique for picking an optimal k Assign τ j Gürkaynak et al. (2007, 2010) estimate both β and τ in the extended Nelson-Siegel simultaneously through a (unconstrained) nonlinear minimization. According to their empirical 335 report, computational efficiency matters in nonlinear optimization algorithms. For the multiple exponential decay model, k terms of Ψ j (m) imply a total number of (2k 1) parameters to be pinned down. Direct estimation in a lump-sum style is not preferable if k is large The maturity range of 0-1 years in the real Treasury market is not available in that less-than-one-year TIPS securities are excluded from the data set. AppendixA explains the rationale. 10 I manage to replicate the estimation method in Gürkaynak et al. (2007). It takes more than one hour in MATLAB to obtain a locally optimal solution with k 5. This is definitely not practical if we want to estimate both real and nominal yield curves for each of the last 201 months, let alone financial practitioners who want to use an interpolation model to refit yield curves with high frequency data. 15

16 This fact becomes a motivation to modify the existing estimation method recommended by the literature. The idea of the alternative approach is rooted in the characteristics of the exponentialbased interpolation function. Section 3 concludes that τ regulates the rate of decay for each exponential basis function. Different values of τ thus depict curvatures at different maturities. The distribution of outstanding maturities in the U.S. Treasury market is known to be skew to the short horizon: in general there are more short, less medium, and even sparser long maturities outstanding. This arguably follows that there are richer shapes clustering along short- and medium-term. E-fold Life then suggests that a larger portion of individual τ s (τ {τ j } k j=1 ) should be given small values. This logic entitles the following procedure to pin down parameters {τ j } k j=1 : instead of estimating them directly, we can assign a value for τ j based on the distribution of maturities outstanding, and then estimate β j accordingly. Specifically, given k and a total of n maturities, m {m i } n i=1, the assignment process can be realized as follows: Step 1: Let τ k so that Ψ k (m) = m, and β k hence governs the level of f(m) at infinity (m ); 355 Step 2: Given 2 k n, mark each k-percentile along the distribution of {m i } n } i=1, {m l l = ñj/k, j = 1,, k 1, where ñj/k is the round-down integer of (nj/k) (any tilde represents corresponding round-down integer and henceforth); Step 3: Assign {τ j = m l j = 1,, k 1, l = ñj/k } Estimate β j The paper introduces two methods to estimate parameter β {β j } k 360 j=1 given that k and {τ j } k j=1 are predetermined. The first method is Nonlinear Minimization, in which β is optimally chosen by minimizing the weighted sum of the squared deviations between the actual prices of Treasury securities and the estimated ones. The weight is the inverse of the bid-asked price spread of each individual security 11. This method again involves nonlinear 11 Securities with a larger bid-asked price spread are most likely those off-the-run securities with lower liquidity, and hence give less information about the market, vice versa. 16

17 optimization algorithms 12, which are time-inefficient and sometimes can not return globally 365 optimal solutions by, for example, the basic optimization toolbox in MATLAB. A natural way to solve this problem is to transform the nonlinear problem to a linear one and apply least squares directly. The only difficulty is the missing of coupon price quotations; however, coupon payment values can be numerically solved out by an iterative procedure. Following McCulloch and Kochin (2000), I introduce the second estimation 370 method Iterative Linear Least Squares (ILLS, henceforth). Briefly speaking, we obtain a first guess of β by each security s yield-to-maturity and then estimate β iteratively through weighted least squares. The coupon value in each iteration is computed by the estimated value ˆβ from the last iteration. AppendixB gives details of the two methods. Although either of the two methods will return an optimal solution for β, ILLS has 375 significant advantages over Nonlinear Minimization in terms of computational efficiency. As an illustration, Figure 2 shows the estimated real and nominal U.S. (negative) log discount function φ(m) on a randomly picked date, February 26, , based on both methods. In Figure 2, the cross signs give the derived log prices net of coupon values based on actual data. The solid curves are fitted curves for the negative log discount function. Notice 380 that they are two (almost) overlapped curves fitted by respective estimation method, which suggests the equivalent effectiveness of the two methods. ILLS however takes significantly less computational time (showed in the legend boxes). I hence recommend ILLS for the yield curve estimation and use it to report the empirical results in the remainder of this section 14. [Figure 2 about here.] 385 Based on the fitted log discount functions, Figure 3 depicts the real U.S. term structure on February 26, The upper left graph in Figure 3 is associated with the real log discount function derived by the multiple exponential decay model in Figure 2. The other two are 12 Although β j is linear in log discount function, the discount function will become exponential in β j according to Eq. 6 and enter the price function through Eq. 1. The objective function will thus be nonlinear and complicated. 13 On that business day, 31 TIPS bonds and 198 T-Notes & Bonds were outstanding. According to the data pre-processing procedures in AppendixA, 29 TIPS and 142 T-Notes & Bonds are pre-selected to estimate β. 14 The quantitative comparison conducted in Section 3.4 is also based on ILLS. 17

18 given by the Semi-natural Cubic Spline and the extended Nelson-Siegel for a comparison [Figure 3 about here.] There are three findings obtained from Figure 3. First, the pattern of the long horizon decay differs between the cubic spline model and models based on exponential functions. As expected, the curves generated by the spline-based model approach to a constant beyond the longest maturity outstanding. In contrast, the multiple exponential decay model and the extended Nelson-Siegel feature an asymptotical decay to infinity in the forward rate curve and another asymptote in the spot yield curves. Yet yield curves driven by different exponential-based models do not share an identical asymptote in that the estimate of β j associated with the common term Ψ j (m) = m differs between models 15. Second, although all three models generate a hump-shape in the forward rate curve, the curve driven by the spline-based model reaches a peak in 8-year while the other two in 12-year approximately. Moreover, the forward rate curve in the cubic spline model is steeper and has more subtle curvatures. Third, even though the two exponential-based models generate similar yield curves, the multiple exponential decay model has three terms in its model specification (k = 3) while the extended Nelson-Siegel has four. The multiple exponential decay model also has one less term than the cubic spline model in this case. It suggests that the multiple exponential decay model could be able to use the most parsimonious model specification to return equally satisfactory yield curves. For all the results showed in Figure 3, I pick k = 3 for the multiple exponential decay model and k = 4 for the cubic spline model. In next subsection, I will explain the principle to pin down this parameter optimally Pin down k 415 Parameter k determines the number of terms, or the number of parameters β and τ, in a nonparametric interpolation model. A model with one term only, for instance, fits just a straight line, whose fitting error with respect to an actual yield curve is likely to exhibit positive serial correlation. On the other hand, too many terms will unnecessarily bring 15 Section 3 suggests that that β j actually governs the long-horizon level of convergence. 18

19 excessive shapes to the fitted curves, which may in turn cause negative serial correlation. In reality, different days will give different shapes of yield curves: some are pretty flat, some are upward-sloping, and still some have more curvatures or humps. It thus may hurt to have either redundant or inadequate parameters in an interpolation model. Moreover, we always want a parsimonious model expression to fit yield curves at a certain point of time. 420 Therefore, it is necessary to find out how many parameters/terms are needed at least in the target function in order to satisfy: (1) the cumulative error will be minimum (overall goodness of fit); (2) error terms should be white noise, without showing any serial correlation. D-W test is one of the candidates to test serial correlation in an error term. Figure 4 shows D-W statistics obtained through testing error terms of the fitted U.S. term structure of 425 real interest rates on February 26, The results are derived by the multiple exponential decay model and the semi-natural cubic spline model respectively for a comparison. Two sets of D-W statistics values are hence obtained corresponding to two different models, of which the multiple exponential decay model is demonstrated on the left figure of Figure 4 and the other model is on the right. Specifically, I obtain a D-W statistics value for each 430 k (as showed along the horizontal axis) in an ascending order from k = 2 to the one whose implied D-W statistics exceeds its corresponding upper critical value. The 95% confidence interval and the mean critical value 16 are numerically simulated under the given regressor matrix Ψ(m) for each value of k. They are depicted respectively on the graph. As seen from Figure 4, the values of the D-W statistics has an upward-sloping trend as k increases. 435 The smallest k whose D-W statistic exceeds its corresponding lower critical value gives the most parsimonious model specification free from serial correlation (at 5% significant level) in the interpolation error term. I call it an acceptable realization of k. One can also pick the k whose D-W statistic is closest to its mean critical value. This k ensures the lowest probability in terms of serial correlation, but may lead to a less parsimonious model. For 440 example, k = 3 for the multiple exponential decay model happens to be an acceptable k as well as the k least likely to have serial correlation on February 26, [Figure 4 about here.] 16 Lower, mean, and upper critical values are at 5% significant level and based on one-tail test. 19

20 However, D-W test assumes a normal distribution in the error term and is not appropriate for higher-order autocorrelation. Now that samples are not necessarily drew from a same normal distribution, D-W test may not be applicable for any data set. Another candidate for the serial correlation test is the nonparametric runs test. It tests the mutual independence of a two-valued sequence, with the null hypothesis that elements in the sequence come in random order against the alternative that they do not. Specifically, a run of an error sequence in our problem is a maximal non-empty segment of the sequence consisting of consecutive elements above or below zero. Figure 5 shows the P-values of runs test with respect to each k assigned for the real U.S. term structures on February 26, Similar to D-W test, the smallest k giving a P-value above 5% in the runs test is just acceptable and leads to the most parsimonious model, while the k associated with a P-value closest to 1 is least likely to have autocorrelation. The former k for the multiple exponential decay model in Figure 5 is equal to 3, while the latter is 4 or 5. In addition, any k smaller than 3 or larger than 6 would cause an autocorrelated error term under the multiple exponential decay model. [Figure 5 about here.] As for overall goodness of fit, I choose the Bayesian information criterion (BIC) for model selection 17. Figure 6 shows BIC values for each value of k specific to the nonparametric model specification of two different interpolation models respectively in fitting the real U.S. yield curves on February 26, In terms of the multiple exponential decay model, k = 3 turns out to be the best overall fit because of the lowest BIC value associated. This value of k coincides with the k recommended by the serial correlation test. I hence pin down k = 3 for the real yield curve-fitting based on the multiple exponential decay model in Section 4.2. [Figure 6 about here.] I repeat these tests on both the real and nominal U.S. Treasury markets for several months in recent years and find a stylized fact that k n, where n is the number of ( ) ˆε ˆε 17 By assuming white noise fitting errors, the BIC takes the Gaussian special form: BIC = n log + n k log (n) 20

21 effective securities. This rule can help pin down k more efficiently and can be adjusted 470 seasonally. 5. Constrained Estimation: An Extension Macroeconomists use the estimated yield curves to interpret monetary policies or business cycles over a period of time. Financial practitioners may instead need them to look for pricing anomalies. Is the proposed interpolation model as well as the estimation method able to 475 coordinate both needs? The answer is positive if the estimation methods for β in Section 4.2 are modified. Three solutions are introduced to help financial practitioners find instantaneous investment opportunities in Treasury markets. All three are based on the same idea: bounding high/low (outstanding) yields gives high/low (estimated) yield curves so that the over- 480 /under-priced securities are screened out from the pool. Treasury security buyers may thus keep an eye on a high yield curve to pick those underpriced securities only, and sellers are interested in a low yield curve instead. Because estimation methods in this section largely involve price constraints, I call them constrained estimation while the estimation in last section, either nonlinear minimization or ILLS, unconstrained estimation. Without loss of 485 generality, I illustrate all constrained estimation methods by bounding high yields or underpriced securities measured by asked prices. The first constrained estimation method, Selected Fit, actually follows the idea of unconstrained estimation. Specifically, I run an unconstrained estimation first using all effective securities to obtain estimated prices ˆp; then select security i such that p i ˆp i and run the 490 unconstrained estimation again using these underpriced securities only. Though being easy and straightforward, this method ignores a portion of market information and thus results in estimation biasness. The second and third constrained estimation methods are the revision to the two unconstrained estimation methods, i.e. nonlinear minimization and ILLS. The revised nonlinear 495 minimization is a constrained minimization by imposing a constraint of nonnegative price deviation. For the revised ILLS I turn to quadratic programming by minimizing the sum of the squared log net price deviations subject to a nonnegative constraint. Similar to ILLS, I 21

22 500 then iteratively estimate β until certain termination condition is met. I call this estimation Iterative Constrained Quadratic Programming (ICQP, henceforth). AppendixB gives the details of all constrained estimation methods. [Figure 7 about here.] 505 Figure 7 shows three sets of fitted real yield curves based on the same data set and interpolation model, but estimated respectively by ILLS, Selected Fit, and ICQP. I recommend ICQP to financial practitioners in their investment decisions because (1) ICQP outperforms the constrained nonlinear minimization in terms of computational efficiency; and (2) Selected Fit is less favorable in terms of the estimation biasness. 6. Conclusion Generally, three contributions are present in this paper. First, this paper proposes a new interpolation model, multiple exponential decay model, to fit yield curves of both real and nominal U.S. Treasury markets. Compared with alternative interpolation models (either spline-based or exponential-based interpolation), this new model is more parsimonious in its model specification, more adaptive to a variety of shapes associated with yield curves, and more efficient to smooth through idiosyncratic variations generated by individual securities. Second, this paper introduces Constrained Estimation, a series of new estimation methods to help look for price anomalies. Curve fitting through the unconstrained estimation is also exclusively discussed and compared. Third and also the most unique contribution is a new procedure implemented to optimally adjust the exact functional form. The functional form of a nonparametric interpolation model is adjustable to specific data set. In this paper, I turn to BIC for overall goodness of fit and D-W test as well as Runs test for randomness in error terms. These statistical tools enable us to take into account validity, optimality, and parsimoniousness simultaneously in specifying a model. In addition, I extend QN cubic spline model and compare its fitted yield curves with those of the proposed multiple exponential decay model. A horse race among the competing models concludes that the multiple exponential decay model outperforms alternative models and is hence able to give the best fit. 22

23 Acknowledgments The author is indebted to J. Huston McCulloch, Paul Evans, Pok-Sang Lam, Yangru Wu, Jimmy Ran, seminar participants at CEF2011, The Ohio State University, Central University of Finance and Economics, Shanghai University of Finance and Economics, Southwestern 530 University of Finance and Economics, and anonymous referees for helpful comments and suggestions. Any remaining errors are the author s responsibility. AppendixA. Data Preprocessing According to conventions, the U.S. real yield curves are derived from TIPS market quotes, while the nominal counterparts are driven by T-Bills, -Notes, and -Bonds. Treasury market 535 quotes are daily archived by common sources such as Bloomberg. The data sets are reasonably equivalent to the proprietary database used by Gürkaynak et al. (2007). I recommend the public database because it is more available to practitioners. Prior to estimation, it is necessary to preprocess the original data. First, I convert all quoted clean prices to dirty prices by taking into account accrued interests. Then all quoted semiannually compounded yield-to-maturities B are converted to their continuously compounded counterparts R by: ( R = 2 log 1 + B ). 2 The remaining work is to select effective securities for estimation. For TIPS, I exclude less-than-one-year TIPSs from the data set. This is because TIPS has been indexed by past 540 inflation instead of future inflation with an approximately 2.5-month s indexation lag 18. It is well known that there is seasonal fluctuation in CPI-U 19, and hence a significant seasonal 18 The inflation adjustment to TIPS principal is daily-based. The reference CPI for each day of a new month is determined by the Treasury through a linear interpolation in the mid of the current month as soon as the CPI-U for the last month is released by Bureau of Labor Statistics. Specifically, on the d t th day of the month ( with a total of d n days, the reference CPI (CP Id r t ) is given by CP Id r d t = CP I t 1 ( 2) d n + CP I ( 3) 1 dt 1 d n ), where CP I ( 2) and CP I ( 3) are the 2 and 3 months lagged CPI-U respectively. In effect, this gives the TIPS an indexation lag of approximately 2.5 months 19 CPI-U, the non-seasonally adjusted U.S. City Average All Items Consumer Price Index for All Urban Consumers, is the specific CPI index that the principal of TIPS is tied to. 23