Estimating yield curves of the U.S. Treasury securities: An interpolation approach

• Date: 01 April 2019
• Published date: 01 April 2019
• DOI: http://doi.org/10.1002/rfe.1039
• Author: Feng Guo

ORIGINAL ARTICLE

Estimating yield curves of the U.S. Treasury securities: An

interpolation approach

Feng Guo

Chinese Academy of Finance and

Development, Central University of

Finance and Economics, Beijing, China

100081

Correspondence

Feng Guo, Chinese Academy of Finance

and Development, Central University of

Finance and Economics, 39 South College

Road, Beijing 100081, China.

Email: guofengkevin@cufe.edu.cn

Abstract

Following the approach of interpolation, this paper proposes the multiple expo-

nential decay model to fit yield curves for both the U.S. TIPS market and the

conventional Treasury security market. Several estimation methods, including the

unconstrained/constrained nonlinear minimization, quadratic programming, and

the iterative linear least squares, are applied to estimate the unknown parameters

according to different curve-fitting purposes. Comparisons between the proposed

model and the alternatives show that the multiple exponential decay successfully

(1) adapts to a variety of shapes associated with yield curves, (2) (partially) keeps

in line with the economic interpretations of Nelson–Siegel summarized by Die-

bold and Li (2006), and (3) dominates the competing models in curve-fitting per-

formance measured by mean fitted-price errors over the sample period. In

addition, the exact specification of a nonparametric interpolation model is pinned

down by applying three statistical tools, which enable us to jointly take into

account validity, optimality, and parsimoniousness of the proposed model.

JEL CLASSIFICATION

G12, E43, C52

KEYWORDS

interpolation, term structure of interest rates, TIPS, treasury security, yield curve history

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INTRODUCTION

A continuous yield curve over the full spectrum of maturities is of great help to interpret macroeconomic fluctuations, mon-

etary policies, and speculation/hedging behavior. Estimating the yield curves, either in real or nominal terms, is therefore a

good place to start relevant research on those topics. “Yield curve”here refers to the discount function, the zero-coupon

yield, the par yield, or the forward rate since each is a transformation of the others. Basically, they are all based on the

benchmark yields implied by price quotes in the Treasury security market.

If the Treasury issued bills, notes, and bonds with a continuous spectrum of maturities, then we could simply “observe”

the benchmark yield curves. In practice, however, the U.S. Treasury has instead issued a limited number of securities with

different maturities and coupon rates. Each of these can be viewed as a basket of zero-coupon securities: one for the princi-

pal payment at maturity and the rest for coupon payments at each semiannual coupon date. 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”

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”yields. One is featured by applying cross-

sectional 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 Kochin

Received: 19 May 2018

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Accepted: 5 July 2018

DOI: 10.1002/rfe.1039

Rev Financ Econ. 2019;37:297–321. wileyonlinelibrary.com/journal/rfe © 2018 The University of New Orleans

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(2000), Nelson and Siegel (1987), and Fisher, Nychka and Zervos (1995) follow this line, although these studi es differ con-

siderably in respective interpolation models and estimation techniques. The other approach in turn focuses on fully speci-

fied, dynamic term structure models. In that line of research, a prevailing approach is to build up reduced-form term

structure models in which yields are expressed as a constant-plus-linear function of underlying state variable(s); both of the

(latent) factors and their factor loadings follow dynamic stochastic processes. This kind of term structure models is well-

known as affine term structure models (ATSM), evolved through the one-factor models such as Vasicek (1977) and Cox,

Ingersoll and Ross (1985), and established/refined by Duffie and Kan (1996), Dai and Singleton (2000), Ang and Piazzesi

(2003), etc. A successful element of ATSM is the cross-asset restrictions imposed on the model to eliminate arbitrage

opportunities. However, the poor empirical performance of the canonical ATSM is often criticized in the literat ure (Chris-

tensen, Diebold & Rudebusch, 2011).

Despite the popularity of the dynamic term structure models, a number of desirable properties make the static approach

—curve-fitting methods, irreplaceable in empirical research as well as real application. First, interpolation models are inde-

pendent of any stochastic processes or equilibrium settings. This type of models is pure descriptive in that the yiel d is a

function of term to maturity only. Second, this approach allows simple, parsimonious functional form s to represent a wide

range of shapes generally associated with yield curves. Nelson and Siegel (1987) advocate parsimonious yield curve models

in that “the whole term structure of yields can be described more compactly by a few parameters.”Third, the simple func-

tional forms, supported by appropriate estimation techniques, substantially reduce computational burden and are also easier

to be understood and applied. This single advantage makes curve-fitting methods widely accepted among practitioners and

central bank analysts (Alfaro, 2011).

An interpolation typically starts with specifying a functional form either to approximate a discount function or a forward

rate function, and then estimates the unknown parameters of the specification. An extensive body of literature hence con-

tributes to develop favorable functional forms as well as efficient estimation techniques. Among them, two popular

approaches exhibit great potential to facilitate high frequency fit: one is McCulloch cubic spline (McCulloch & Kochin,

2000) and the other is the extended Nelson–Siegel (Nelson & Siegel, 1987; Svensson, 1994). The former proposes a Quad-

ratic-Natural (QN, henceforth) cubic spline to fit a negative log discount function, whose parameters are estimated by an

iterative method based upon linear least squares. The latter instead is established upon exponential decay functions and

based on nonlinear minimization methods. Moreover, the latter has been successfully extended to dynamic Nelson–Siegel

by Diebold and Li (2006), to adaptive dynamic Nelson–Siegel by Chen and Niu (2014), and further to the affine arbitrage-

free class of Nelson–Siegel by Christensen et al. (2011). These researchers , and still others (Coroneo, Nyholm & Vidova-

Koleva, 2011), validate that the Nelson–Siegel class has economically intuitive properties as yield curve factors coincide

with the modern three factors: level, slope, and curvature. Gürkaynak, Sack and Wright (2007), however, point out that nei-

ther of the two approaches wins over the other; the choice between the two largely depends on the purpose of yield curve

fitting. They argue that the two interpolation models differ in flexibility each allows the fitted yield curves to 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. On the contrary, macroeconomists may prefer the

more 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

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further research managing to integrate the strengths of the two approaches and put forward new static interpolation models,

which may in turn shed light on the discovery of a new class of ATSM. 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, which target function 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 the

economic background in theory as well as the practicability in real applications? Third, how do we estimate the parameters

in the proposed model to realize the best computational efficiency as well as robustness?

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, which makes the estimation largely easier. However

the behavior of the fitted yield curve at long-end becomes hard to govern. This problem is solved by another approach: the

bond pricing function can be specified in terms of the negative logarithm discount function with properly added end-point

conditions. Nevertheless, the consequent nonlinear pricing function complicates model estimation. McCulloch and Koch in

(2000) actually follow this approach.

The second dimension is to specify an interpolation functional form for the target function selec ted. Again, two

approaches are competing in this area. The QN cubic spline approach, introduced by McCulloch and Kochin (2000), fea-

tures a nonparametric model specification. It allows substantial flexibility on the shape of yield curves, but we do not need

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