Discrepancies in the underlying zero-coupon yield curve
Antonio Díaz *
Francisco Jareño *
Eliseo Navarro **
* Departamento de Análisis Económico y Finanzas, Universidad de Castilla-La Mancha
Facultad C. Económicas y Empresariales, Plaza de la Universidad, 1, 02071 Albacete (Spain)
Antonio.Diaz@uclm.es ; Francisco.Jareno@uclm.es ; Tel: +34 967599200
** Departamento de Economía y Dirección de Empresas, Universidad de Alcalá
Edificio C. Económicas, Empresariales y Turismo, Pz. de la Victoria, s/n, 28802 Alcalá de Henares (Spain)
Eliseo.Navarro@uah.es , Tel: +34 918854295
THIS DRAFT: September 2014
The financial literature and financial industry need zero-coupon yield curves as an input for testing
hypotheses, pricing assets, and managing risk. The final users of such yield curves assume that the
available datasets are accurate, but data providers often use different sample selection criteria and
methodologies to estimate their yield curves. Considering three popular interest rate datasets (from the
Federal Reserve Board, the US Department of the Treasury, and Bloomberg), we explore the extent to
which the selected dataset may determine the results of different analyses. To examine the implications
of the choice of dataset, we estimate our own daily yield curves from four alternative asset baskets from
GovPX market data, which mimic those of each data provider. We study the properties of the raw yield
curves and their impact on both volatilities and the expectations hypothesis tests. We provide guidance
to final users for selecting a yield curve dataset.
Keywords: Term Structure of Interest Rates; Yield Curve Datasets; Volatility Term Structure; Forward
rates; Expectations Hypothesis
JEL Classification: E43; F31; G12; G13; G15
Acknowledgements: We are indebted to Zvika Afik, Alain Coën, Manfred Frühwirth, Alois Geyer,
Robert Korajczyk, John Merrick, Pietro Millosovich, Alfonso Novales, Christian Speck, Christian
Wagner, Josef Zechner and Hairui Zhang for very detailed suggestions.
1.- Introduction
The term structure of interest rates or yield curves (YCs) is a basic input for most
practitioners and researchers. These zero-coupon interest rates are tremendously important for
a number of purposes in finance, monetary policy, economic theory, and so forth.1 The final
users of YCs usually download this information from a database provider. Among such users, a
few YC datasets have become popular, some because they are publically available and free and
others because they are offered by the primary financial data providers.
One of the sources of divergence between YC datasets is the fitting method. Examining and
comparing the trade-off between the goodness of fit and the smoothness of different fitting
techniques to obtain the best fit for their own YCs from market data are beyond the scope of the
vast majority of analyses. Even specific studies focused on such comparisons obtain ambiguous
results. For instance, both Bliss (1996) and Bolder and Gusba (2002) note the trade-off between
the closeness of the fit to the set of observed government coupon prices and the smoothness of
the corresponding zero-coupon rate function. As result of the horse races, they propose winners
and losers. Bliss (1996) examines five models and proposes a winner (the unsmoothed FamaBliss) and a loser (the Fisher-Nychka-Zervos). Paradoxically, Bolder and Gusba (2002)
highlight two winners among eight candidates, i.e., an exponential version of the Merrill Lynch
exponential spline and the Fisher-Nychka-Zervos from the zero-coupon curve.
Most papers in the literature and most practitioners appear to be unconcerned about the
implications of the key decisions that database providers make in estimating their YCs. The
popular YC datasets alternatively consider transaction prices, quoted prices, or market yields to
maturity for different sets of risk-free debt instruments, including/excluding bills, on-the-run
bonds, callable bonds, and so forth, and they use different models and numerical techniques to
estimate the zero-coupon interest rates. Researchers and practitioners often assume that one of
these available YC datasets is perfectly accurate for their purposes without further
consideration. A question that we would like to answer is, to what extent are the results that
these final users obtain from these YCs conditioned or even “contaminated” by the particular set
of zero-coupon interest rate data that is employed? Does the fitting method or the bond sample
that we (or someone else) use to estimate the term structure of interest rates matter for the
result of these studies? Which of the existing YCs fits best to answer a specific economic
question?
This study investigates the possible implications of considering one among several
alternative commonly accepted YC datasets. We would like to answer the question, what is the
best YC dataset to use? Of course, this question has no answer or, at least, no simple and clear
answer. The answer depends on the purpose. Final users of YCs address different issues and
have different goals. Some of these users demand flexible YCs that fit as well as possible in
terms of correctly pricing existing assets. For instance, they may use YCs to manage interest
rate risk, to calibrate fixed income valuation models, to hedge interest rate exposures, to
provide a benchmark for pricing other assets, and so forth. For these users, it is also relevant
whether the YC was estimated from on-the-run government bonds (i.e., the most liquid bonds)
or shows the average market liquidity from off-the-run bonds (i.e., bonds with “regular”
liquidity). Other users, however, demand YCs that are rigid and stable over time. Such YCs are
required in research on macroeconomic contexts, monetary policy, and investors’ risk
preferences. Based on our analysis, we propose recommendations and highlight concerns about
the use of each one of the considered YC datasets. We try to provide guidance to the community
about which dataset to use for a specific problem.
As examples of these purposes, we could mention pricing bonds, pricing fixed-income derivative products,
constructing forward interest rates, determining risk premia, testing the expectations hypothesis, examining the
ability of volatility to capture economic uncertainty, analyzing the volatility transmission along the YC, forecasting
excess returns and interest rates, measuring risk, designing hedging strategies, and so forth.
1
1
We examine daily yield curves from the Federal Reserve Board (FRB in advance),2 the U.S.
Department of Treasury (DoT), and Bloomberg during the sample period January 1994 to
December 2006. We get the datasets from Treasury YC estimates of the FRB posted on its
website and commented by Gürkaynak et al. (2007), from the YC reported by the DoT,3 and
from the Bloomberg (F082) zero coupon yield curve. We omit other common YC dataset on
research papers, the unsmoothed Fama and Bliss (1987) dataset.4 It is monthly and only
provides estimates out to five-year maturities. Maturity spectrum is a key determinant of the
YC estimate properties. This dataset avoids the main difficulty to fit the YC, i.e. dealing with
the convexity effect originated by securities with maturities of twenty years or more.
We examine daily YCs from the Federal Reserve Board (FRB),5 the US Department of the
Treasury (DoT), and Bloomberg during the sample period of January 1994 to December 2006.
We obtain the datasets from Treasury YC estimates of the FRB posted on its website and noted
by Gürkaynak et al. (2007), from YCs reported by the DoT,6 and from the Bloomberg (F082)
zero-coupon YC. We omit other common YC dataset in research papers, i.e., the unsmoothed
Fama and Bliss (1987) dataset,7 which is monthly and provides estimates out to five-year
maturities only. The maturity spectrum is a key determinant of the estimate properties of YCs.
This dataset thus avoids the primary difficulty of fitting YCs, i.e., addressing the convexity
effect originated by securities with maturities of twenty years or more.
The providers of the three considered datasets use different fitting techniques, different
market data, and different baskets of assets. There are several reasons underlying their model
and data choices. In the case of the FRB, Gürkaynak et al. (2007) provide a fully detailed
description. Information about the DoT and Bloomberg datasets is scarce, however. The
different estimation methods are a weighted Svensson (1994) model (FRB), a quasi-cubic
hermite spline function (DoT), and a piecewise linear function (F082). Further, the providers
analyze different basket sets of securities: off-the-run bonds (FRB) and on-the-run bills and
bonds (DoT). Finally, they consider different market data as an input: market prices (FRB),
market yields (DoT), and generic prices (F082).
We propose an indirect method to compare the characteristics of these three datasets to
propose recommendations, as a direct comparison is not possible. Neither the prices from which
they fit YCs nor the detailed fitting method of two providers are available to us. Thus, we
cannot directly analyze the in-sample and out-sample goodness of fit. As an indirect method, we
try to mimic the composition of the basket of assets that is used as the input by each YC
provider. We form four different samples of assets: sample A, which constitutes the full sample
(bills and straight bonds); sample B, which excludes the on-the-run and first off-the-run assets
of each maturity; sample C, which also excludes bills and approximately mimics the FRB sample
composition; and sample D, which mimics the DoT sample composition (eleven maturities of
on-the-run securities). From the market prices for each sample of assets reported by GovPX, we
This dataset is called “the FRB dataset” in this paper for convenience. However, the spreadsheet that can be
downloaded from the FRB website contains this sentence: “Note: This is not an official Federal Reserve statistical
release.”
It
is
usually
cited
in
the
literature
as
Federal
Reserve
H15
series.
http://www.federalreserve.gov/econresdata/researchdata.htm
3 http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
4 An iterative method of forward extraction is used from month-end price quotes obtained from the CRSP
Government Bonds files. The unsmoothed Fama-Bliss discount rates are often “smoothed” using an extended
Nelson-Siegel method.
5 This dataset is called “the FRB dataset” in this paper for convenience. However, the spreadsheet that can be
downloaded from the FRB website contains this sentence: “Note: This is not an official Federal Reserve statistical
release.” It is usually cited in the literature as the Federal Reserve H15 series.
http://www.federalreserve.gov/econresdata/researchdata.htm
6 http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/TextView.aspx?data=yield
7 An iterative method of forward extraction is used from month-end price quotes that are obtained from the CRSP
Government Bond files. The unsmoothed Fama-Bliss discount rates are often “smoothed” by using an extended
Nelson-Siegel method.
2
2
fit our own YC by using both the unweighted and the weighted versions of the Svensson
method.
Our empirical analysis consists of three stages. In the first stage, we examine the properties
of the raw YC. We observe the effects of the YC dataset on a number of aspects, such as the
level, shape, temporal dynamics, and forward rate correlation. We separately analyze the model
choice and the data choice decisions of the YC providers on the resulting spot rates. Then, we
examine the impact of different fitting methods from the same dataset. We observe that a simple
assumption about the structure of the variance of the error term, as an example of difference in
the fitting method, affects not only the quality of the adjustment but also the results of further
analyses based on these YCs. Finally, we study the impact of different datasets from the same
fitting method on the resulting YCs. From this analysis, we observe a possible mispricing
problem in the short end of the FRB YC because it does not consider bills. However, this YC
provides a good fit for the rest of the maturity spectrum, with extra-good fit for the 1-year
maturity in periods of flights to liquidity. The DoT sample composition guarantees a good fit
for short maturities, whereas maturities longer than 10 years are clearly neglected. Both the
DoT and the F082 YCs are far from being smooth curves. This lack of smoothness can generate
serious inconsistencies in the forward rate term structure. Finally, the three external datasets
show a clear convexity problem in the long end of the YCs, as reported by Gürkaynak et al.
(2007).
As a second stage, we consider the impact of each YC dataset in the resulting volatility term
structure (VTS). We estimate historical volatilities and EGARCH volatilities from the three YC
datasets and our own estimations. The estimates from the three external YC datasets show
lower levels of volatility in the short end of the YCs (maturities shorter than 1 year) than our
estimations. The F082 YC presents high volatility for maturities longer than 5 years. Finally,
excluding on-the-run securities reduces the volatility for the entire maturity spectrum.
The third stage examines the impact of the different YC datasets on the expectations
hypothesis tests. We study which YC dataset provides results most consistent with this theory.
Rather than aiming to find evidence supporting or rejecting the theory, we rank the considered
YC datasets by their ability to approach the theory. We obtain results that apparently support
the expectations hypothesis in the short end of the YC. The best properties are provided by the
DoT YC. The results for maturities longer than 1 year do not appear to depend on the model
and data choices of each YC dataset.
Our analysis contributes to the literature in several ways. First, we compare three of the
most popular YC datasets among researches and practitioners and highlight the primary
differences between them with respect to the fitting method, sample composition, and market
input (prices/yields). Second, we propose an indirect method to analyze the consequences of the
key decisions of each data provider in composing their dataset. We then examine the
performance of a common fitting method applied to baskets of assets that mimic each dataset
composition from our market prices dataset. Third, we highlight the relevance of the fitting
technique by adjusting two versions of one of the most popular approaches, the Svensson (1994)
model, applied to the same basket of assets. Fourth, we examine the impact of using a
determined YC dataset to answer several economic problems, such as determining the VTS or
testing the expectations hypothesis. Fifth, we provide guidance for selecting a YC dataset based
on the addressed economic application.
The rest of this paper is organized as follows: Section 2 describes the YC datasets. The first
part of this section analyzes the primary characteristics of the external YC data that we
examine. The second part of the section then describes the fitting process for our own estimates
using the Svensson method. For this purpose, we describe the original dataset that we use, the
model, and the assumption about the error term variance. In section 3, we perform an empirical
analysis that consists of a study of the impact of the methodology for estimating spot rates on
YCs, the impact on the term structure of volatilities, and the impact on the expectations
3
hypothesis tests. The last section provides recommendations from the summary of results and
includes the conclusions.
2.- The alternative yield curve datasets
2.1- Popular zero yield curve datasets
We compile daily zero-coupon YCs for the US Treasury market from three popular
external datasets, i.e., the FRB, the DoT, and F082, for the period from January 1994 to
December 2006. These YCs are calculated by using different estimation methods, security sets,
and bond prices. Daily interest rates for the full spectrum of maturities up to 30 years are
computed from these YC datasets.
The first YC dataset that we examine comprises the Treasury YC estimates of the FRB
posted on its website and noted by Gürkaynak et al. (2007). For this dataset, the FRB uses a
weighted version of the Svensson (1994) method from prices of all of the outstanding off-therun bonds. Among other securities, they exclude from their estimation all Treasury bills and the
on-the-run and “first-off-the-run” issues of bonds and notes.
We also analyze the YCs reported by the US DoT. The Treasury does not publish
historical data for these interest rates, but they can be downloaded as H.15 in the Federal
Reserve Statistical Release. They use a quasi-cubic hermite spline function that passes exactly
through the yields for the chosen securities as the YC estimation method.8 Therefore, the DoT
does not estimate a zero-coupon term structure because they merely obtain a YC (a relationship
between yields to maturity and terms to maturity). They consider the curve to be a “par curve”
because the on-the-run securities typically trade close to par. No details about the used
functions are reported by the DoT. As inputs, they use the market yields to maturity for the onthe-run securities.9 They include four maturities for the most recently auctioned bills (4-, 13-,
26-, and 52-weeks), six maturities of just-issued bonds and notes (2-, 3-, 5-, 7-, 10-, and 30years), and the composite rate in the 20-year maturity range.
The last external YC dataset that we consider comprises the Bloomberg (F082) zerocoupon YC, which is estimated by using a piecewise linear function from Bloomberg generic
prices for all of the outstanding Treasury bonds. No details about the used functions are
reported by Bloomberg. They estimate the zero-coupon YC, which they use to generate
“Bloomberg fair value” curves to price most bonds that are traded over-the-counter or that are
illiquid.10
There are several sources of discrepancies among the three external YC datasets. We
highlight three of them: the fitting method, the prices/yields that are used as the input in the
fitting process, and the basket of assets from which the YC is estimated. Table 1 summarizes
some of the characteristics of the datasets.
[INSERT TABLE 1]
First, each data provider uses a different fitting method: a weighted Svensson (1994) model
(FRB), a quasi-cubic hermite spline function (DoT), and a piecewise linear function (F082). We
have not been able to find details about the DoT and F082 fitting methods. Each method has
virtues and flaws. In general, there is a trade-off between having the flexibility to fit complex
http://www.treasury.gov/resource-center/data-chart-center/interest-rates/Pages/yieldmethod.aspx
These market yields are calculated from composites of quotations obtained by the Federal Reserve Bank of New
York.
10 Bloomberg explains that a piecewise model contains more points than a parameterized smooth curve; thus, it
improves the fit of the YC. However, they recognize that this function could also result in unstrippable zero curves
and negative forward rates. See M. Lee, International Bond Market Conference 2007, Taipei.
8
9
4
shapes and having stability and convergence over the long term when comparing methods. The
suitability of each fitting method depends on the purpose that the YC is intended to serve.
Second, the three YC datasets consider different market data as the input. The DoT
employs “close of business” bid yields-to-maturity. Gürkaynak et al. (2007) note that they use
end-of-day prices for their FRB estimates, but they do not specify what type of prices; thus,
prices could be quoted prices (mid bid-ask, bid, or ask prices), the last trading price, arithmetic
average (or weighted average by volume) trading prices over the day, and so forth. Finally,
Bloomberg considers “Bloomberg generic” (BGN) prices, which are computed as the simple
average price of all types of prices, including indicative prices and executable prices, quoted by
their price contributors over a specified time window. Each dataset is constructed by trying to
fit different inputs: most likely quoted prices (FRB), quoted yields to maturity (DoT), and
generic prices (F082). Thus, we could expect to find differences in levels of the resulting YC
estimations of the FRB and F082 datasets.
Another relevant aspect to consider is that the DoT fits yields-to-maturity instead of prices.
As they consider on-the-run bills and bonds, the resulting YC can be interpreted as a par YC
because these just-issued assets are traded near par. We, and most likely other users of the DoT
dataset, make a mistake in using these par YCs as term structures of interest rates. Coupon bias
and forward rate bias imply differences between zero-coupon interest rates and par yields,
especially for long maturities.11
Third, each YC dataset is estimated from different sets of US Treasury securities. The FRB
estimates only include second-off-the-run or older bonds with more than three months to
maturity. Further, they exclude quotes of all securities with less than three months to maturity,
all Treasury bills, bonds with embedded options, twenty-year bonds since 1996, the on-the-run
bond and the first-off-the-run bond for each maturity (2-, 3-, 4-, 5-, 7-, 10-, 20-, and 30-years),
and “other issues that we judgmentally exclude on an ad hoc basis.” The DoT considers the four
most recently auctioned bills (4-, 13-, 26-, and 52-week), the six maturities of on-the-run bonds
and notes (2-, 3-, 5-, 7-, 10-, and 30-years), and the composite rate in the 20-year maturity
range. To fit the F082 series, Bloomberg includes all of the outstanding Treasury bonds, i.e.,
callable or not callable, and traded or not traded during the day. No bills are included.
The basket of securities considered by each of the three YC datasets is quite diverse, and the
different compositions of the dataset of securities from which the YCs are estimated have
multiple implications. The Treasury market includes traded securities with important
differences, such as optionality, tax effects, remaining maturity, and liquidity.
The impact of optionality is clear. During the sample period, a group of old 30-year callable
bonds is outstanding. The prices of these option-bearing bonds depend on the moneyness of the
implicit option. During the sample period, these bonds are traded at extremely high yields to
maturity.12 Only Bloomberg considers these bonds. For most purposes, including these bonds in
the estimation can introduce unnecessary noise and disrupt the necessary homogeneity of the
considered securities.
The Bank of International Settlements (BIS, 2005) technical report emphasizes that premia
induced by tax regulations are notoriously difficult to manage. Several postures can be adopted:
attempting to remove tax premia from the observed prices before they are used in estimations,
excluding instruments with distorted prices from the dataset, or ignoring this problem
altogether. We assume that the three considered YC datasets ignore tax implications.
Coupon bias represents the difference between the yield to maturity of a coupon-bearing bond and the zerocoupon interest rate with the same term to maturity. The yield to maturity for a coupon-bearing bond is a sort of
weighted average of the spot rates associated with the term to maturities of each cash flow. The steeper the term
structure or the larger the coupon is, the greater the coupon bias is. In referring to forward bias, Livingston and
Jain (1982) illustrate the different behavior for long maturities of the implied forward rates obtained from the par
yields and the actual forward rates computed from the zero-coupon interest rates.
12 For the date shown in Figure 3, eleven callable bond issues with remaining maturities between 10 and 15 years
were traded at yields to maturity in the 7.5% to 8.5% range.
11
5
The BIS (2005) comments on the importance of the maturity spectrum. Most central banks
exclude the part of the maturity spectrum for which debt instruments are available. For
instance, certain central banks consider only the interval from one to 10 years. When modelling
the short-end of the term structure, this decision mostly concerns the selection of the most
suitable short-term instrument type and the minimum remaining term to maturity allowed in
the estimation. In this sense, the relevance that the YC datasets assigns to maturities shorter
than one year or longer than ten years markedly differs. Four of the eleven maturities
considered by the DoT fall under the first range (4-, 13-, 26-, and 52-week bills), whereas only
two of them are maturities longer than ten years (20- and 30-year notes). In contrast, the FRB
excludes all bills, regardless of the maturity, and bonds with fewer than three months to
maturity but includes almost all of the straight bonds and notes.13 F082, in turn, includes all
outstanding bonds and notes.
Other aspects related to the maturity spectrum are the number of securities that are
included in the fitting process and the relative number of bonds with the longest maturities.
The fitting from eleven observations in the case of the DoT will provide simple shapes.
However, the fitting from one hundred observations in the case of the FRB,14 or even twice this
number in the case of F082, renders the resulting shapes of the YC much more complex. In
addition, the relative number of long-term bonds is relevant, as the price sensitivity of these
bonds to small interest rate variations is large. The optimization process thus tends to ensure a
fine adjustment in the long end of the YC and devote little attention to the shortest terms to
maturity. Even a weighted error structure is imposed, as explained in section 2.2.3, the shape of
the YC is clearly conditioned by the presence of long-term assets.
Another primary factor underlying different security sample compositions is liquidity.
Liquidity may have an important impact on prices. Sarig and Warga (1989) and Warga (1992)
suggest that younger bonds are usually traded more frequently. Warga (1992) uses an auction
status dummy variable that indicates whether or not an issue is “on-the-run” (i.e., the most
recently issued security of a particular maturity). Amihud and Mendelson (1991) observe that
bonds approaching maturity are significantly less liquid because they are “locked away” in
investors’ portfolios.
Usually when a bond is just issued, it concentrates most of the trading volume because most
investors and fund managers are trying to allocate or distribute this new asset in their
portfolios or within their clients.15 However, when this bond becomes seasoned and, above all,
when new references are issued, the trading volume decreases dramatically, and so does its
liquidity. These stages of a bond’s life have an important impact on prices and yields, as “on-therun” issues often trade at a premium to the remaining outstanding issues. The FRB and the
DoT take opposing positions regarding liquidity. Meanwhile, the DoT considers only “on-therun” issues, whereas the FRB includes only “second-off-the-run” bonds or older.
The above noted sources of discrepancies among the three external YC datasets (i.e., the
fitting method, the prices/yields that are used as the input, and the considered assets) prevent a
direct comparison of the datasets to analyze the trade-off between flexibility and stability and
thus to determine which YC dataset is the most suitable. As an indirect method, we could
calculate pricing errors by using these YCs to replicate prices reported by GovPX. In this sense,
the GovPX dataset provides us with quotes and trades for all US Treasury securities. However,
the results should be inconclusive because our GovPX prices can be quite different from those
used by the FRB, the DoT, and Bloomberg to fit their YCs. Another strategy may be to adjust
13 The
FRB excludes the 30-year on-the-run and first-off-the-run notes.
One hundred observations is approximately the average number of observations in our sample C, which mimics
the FRB security composition (see Table 2).
15 Sack and Elsasser (2004) comment: “Trading volume in nominal Treasury securities was (…) most of this trading took
place in on-the-run issues, which turned over more than fourteen times per week on average and accounted for 74 percent of the
total volume in Treasury coupon securities. (…) Excluding on-the-run issues, we note that the turnover rate for all off-the-run
Treasury coupon securities was approximately 22 percent.”
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6
YCs from the GovPX prices, thereby replicating the fitting methods of the three YC datasets;
however, we do not have sufficient details to replicate the DoT and Bloomberg approaches.
Either way, we highlight the relevance of the fitting technique in the resulting YC by adjusting
two versions of one of the most popular approaches, the Svensson (1994) model, from the
GovPX prices. We apply both the original unweighted error scheme (USV) and the weighted
scheme (WSV) of the Svensson model. The accuracy and stability of the very short-term
interest rates depend largely on this choice of scheme.
Fortunately, the third source of discrepancies is partially under our control. We can fit daily
YCs from baskets of securities that approximately mimic the composition of the security
datasets considered by the FRB and the DoT. We can additionally analyze the consequences of
extending the universe of government securities involved in the estimates by including all of
the traded bills and bonds. In this sense, we construct a different basket of securities from the
prices reported by GovPX to estimate daily YCs with the Svensson method. We examine the
results to propose recommendations and highlight concerns about the use of the three popular
datasets (FRB, DoT, and F082).
2.2.- Indirect comparison method
The three datasets use different market variables (prices/yields) and baskets of assets as
inputs to fit YCs. The “end-of-day prices,” the “close of business” bid yields-to-maturity, and the
“generic” prices (quotes over a time window) used by the FRB, the DoT, and Bloomberg,
respectively, are not available to us. Therefore, we are not able to compute in-sample or out-ofsample performance tests based on fitted-price or fitted-yield errors.
In this section, we propose an indirect comparison method to isolate effects from different
compositions for the basket of assets considered by each method. We generate our own YC
estimates from a common and popular fitting method, the Svensson (1994) approach, applied to
different baskets of assets that try to mimic those used by the YC dataset providers.
Additionally, we propose a simply exercise involving changing an assumption in the Svensson
fitting process to illustrate the impact on the resulting YC of the fitting method.
2.2.1. Our original dataset of government security prices
We obtain intraday US Treasury security quotes and trades for all issues during the period
between January 1994 and December 2006 (2,864 trading days) from the GovPX database.16
GovPX consolidates and posts real-time quotes and trades data from six of the seven major
interdealer brokers (with the notable exception of Cantor Fitzgerald). Collectively, these
brokers account for approximately two-thirds of the voice interdealer broker market. In turn,
the interdealer market is approximately one-half of the total market (see Fleming, 2003). Hence,
while the estimated bills coverage exceeds 90% in every year of the Fleming’s GovPX sample
(Jan 97–Mar 00), the availability of thirty-year bond data is limited because of the prominence of
Cantor Fitzgerald at the long-maturity segment of the market. According to Mizrach and Neely
(2006), voice-brokered trading volume began to decline after 1999 as electronic trading
platforms (e.g., eSpeed, BrokerTec) became available. In fact, GovPX does not provide
aggregate volume and transaction information after May 2001.17 Therefore, we assume an
imperceptible impact from the decline in GovPX market coverage on our estimates because we
consider the midpoint prices and yields between the bid and ask prices at 5 pm.
The GovPX dataset contains snapshots of the market situation at 1 pm, 2 pm, 3 pm, 4 pm,
and 5 pm. Each snapshot includes detailed individual security information, such as CUSIP,
GovPX Inc. was established under the guidance of the Public Securities Association as a joint venture among
voice brokers in 1991 to increase public access to US Treasury security prices.
17 After ICAP’s purchase of GovPX in January 2005, ICAP PLC was the only broker reporting through GovPX.
16
7
coupon, maturity date, and product type (an indicator of whether the security is trading when
issued, on the run, or active off the run). The transaction data include the last trade time, size,
and side (buy or sell), the price (or yield in the case of bills), and the aggregate volume (volume
in millions traded from 6 pm on the previous day to 5 pm). The quote data include the best bid
and ask prices (or discount rate actual/360 in the case of bills) and the mid-price and mid-yield
(actual/365).
Our initial sample relies on the information at 5 pm, i.e., the last transaction occurring
during “regular trading hours” (from 7:30 am to 5:00 pm Eastern Time, ET), if available, and
quote data otherwise. Quote prices are used during the last part of the sample period because
trading volume information is not reported by GovPX. We complement the GovPX data with
official data on the dates for the last issue and the first coupon payment and the coupon rate of
each Treasury security.18 This information is publicly available on the US Treasury website.
To obtain a good adjustment in the short end of YCs, we initially consider all Treasury
bills. In this term to maturity segment, bills are very much more actively traded than old offthe-run notes and bonds.19 Thus, we include only the Treasury notes and bonds that have at
least one year of life remaining. Because the number of outstanding bills with terms to maturity
of between 6 months and 1 year declines considerably during the year 2000 and the 1-year
Treasury bill is no longer auctioned beginning in March 2001, we also consider Treasury notes
and bonds with remaining maturities between 6 and 12 months for the period after 2001.
We also apply other data filters that are designed to enhance the quality of the data. First,
we do not include transactions that are associated with “when-issued” and cash management or
trades and quotes that are related to callable and flower bonds and TIPS (Treasury InflationProtected Securities). Second, when two or more different securities have the same maturity, we
consider only trades and quotes for the youngest security, i.e., the security with the last auction
date. Finally, we exclude yields that differ greatly from yields at nearby maturities.20 For certain
dates, we apply an ad hoc filter. We occasionally observe that deleting a single data point in the
set of prices used to fit the YC can produce a notable shift in both the parameters and the fitted
yields, notably improving the fit. This phenomenon is also noted by Anderson and Sleath
(1999).
Controlling for market conventions, we recalculate the price of each security in a
homogeneous procedure to avoid the effects of different market conventions depending on
maturities and assets. Every price is valued at the trading date on an actual/actual day-count
basis. In the case of Treasury bills, we first obtain the price at the settlement date from the last
trade price, if available, and from the mid-price between the bid and ask prices otherwise.21 In
both cases, the GovPX reported price is a discount rate using the actual/360 basis. Second, we
compute the yield-to-maturity as a compound interest rate by using the actual/actual. Third, we
calculate “our” price at the trading date by using the yield-to-maturity obtained in the previous
step.22 In the case of Treasury notes and bonds, the price is directly reported in the data as the
last trade price or the mid-price. From this price, we apply the mentioned second and third steps
to obtain “our” homogeneous price.23
2.2.2. Mimicking sample compositions
The “standard interest payment” field indirectly provides information to identify callable bonds and TIPS
(Treasury Inflation-Protected Securities). We exclude these assets.
19 In addition, Fleming (2003) emphasizes that GovPX bill coverage is larger than bond and note coverage.
20 These cases include interdealer brokers’ posting errors similar to those mentioned by Fleming (2003).
21 We do not consider the reported mid yield, which is simple interest with an actual/365 basis, except for more
than 6-month remaining maturity bills, which are valued by using the bond equivalent yield.
22 Note that the settlement date is generally a working day after the trading date.
23 We control for the special size of the first interest payment in just-issued securities.
18
8
Four different combinations of government securities are considered as different inputs in
our YC estimates. Table 2 reports detailed information about the composition of our dataset
from GovPx asset prices. “Sample A”—or the “full sample”—refers to the sample that includes
all bills, notes, and bonds that meet the above described requirements. Excluding the on-the-run
and the first off-the-run assets of each maturity from sample A yields “sample B.” Separately,
“sample C” approximately mimics the FRB sample composition by excluding all of the bills and
the on-the-run and first off-the-run securities. Finally, “sample D” mimics the DoT sample
composition, which only includes the on-the-run bills, notes, and bonds for eleven maturities:
four maturities of most recently auctioned bills (4-, 13-, 26-, and 52-weeks), six maturities of
just-issued bonds and notes (2-, 3-, 5-, 7-, 10-, and 30-years), and the composite rate in the 20year maturity range.
[INSERT TABLE 2]
2.2.3. Our term structure specification
As mentioned before, we cannot replicate the fitting methods of two of the considered YC
datasets because we do not have sufficient information about the DoT and Bloomberg
approaches. Thus, we apply the Svensson (1994) method for a number of reasons, which we
explain below. This method is also used for the FRB dataset; however, there are relevant
differences between the methodology that Gürkaynak et al. (2007) employ for this fitting
approach and the methodology that we apply in this paper.
The estimation of the spot rates on a given date consist of finding a functional form that
approximates the theoretical discount function, D(t), and replicating a set of bond prices at a
given instant as accurately as possible:
Pk 
Tnk
C
k
T
·D(T , b)   k
k =1,2, … , m
[1]
T T1k
where Pk is the price of bond k, CTk denote the cash flows (coupon and principal payments)
generated by bond k, D(T, b ) is the discount function that we want to approximate and that
depends on a vector of parameters b , εk is an error term, and m is the number of different assets
included in the estimation.
Practitioners usually estimate YCs from successive swap rates combined with money
market data and/or coupon bond market data. The quoted swap rates can be considered par
yields for bonds, and the simple non-parametric bootstrapping technique can be used to obtain
spot rate estimates for certain fixed maturities. Other maturities are obtained from more or less
sophisticated interpolation methods. In addition, the popular unsmoothed Fama and Bliss
(1987) method is an iterative method of forward rate extraction. The resulting discount rate
function exactly prices the included bonds. From swap rates or from bond prices, the implied
forward curve can be an irregular curve (e.g., with a “sawtooth”) and thus inconsistent and
sensitive to bond price variations/errors.
The term structure can be approximated primarily by using two sets of parametric models
depending on the functional form: models based on the discount function or spline-based
approaches, such as McCulloch (1971), Vasicek and Fong (1982), or Fisher, Nychka, and Zervos
(1995), and models based on the instantaneous forward curve, such as Nelson and Siegel (NS,
1987) or Svensson (1994).
These models can have different degrees of flexibility to describe the term structure of
interest rates (see Bliss, 1996, or Bolder and Gusba, 2002). For instance, some models may have
a greater ability to describe the hump that is so often observed in YCs or the behavior of longterm interest rates. Meanwhile, other models can be more rigid in the adjustment of actual YCs;
9
in this case, some models may produce more or less volatile interest rate estimates in some
tranches of YCs. If a model is too rigid to adapt to the actual shape of a YC, in some regions of
the YC, it may produce estimates of spot rates that fluctuate less than real spot rates. By
contrast, a more flexible model may more adequately capture the real behavior of interest rates.
According to the BIS (2005), nine of thirteen central banks currently use either the NS
(1987) model or the extended version suggested by Svensson (1994) to estimate the term
structure of interest rates.24 One of the exceptions is the United States, which applies a
“smoothing splines” method.
We apply the Svensson (1994) “two-hump” model, which provides us with a more reliable
25
fit. The Svensson (1994) model can be considered an extension of the NS (1985) model. Both
methods are simple parametric models of the term structure of interest rates. These
parsimonious approaches impose a functional form for the instantaneous forward rates. In the
case of the Svensson model, the forward rates are governed by six parameters:
 T
 T
 T
T
T
fT  0  1 exp     2 exp     3 exp   
1
2
 1 
 1 
 2 
[2]
where T is the term to maturity and (β0 , β1 , β2 , τ1 , β3 , τ2) is the set of parameters to be
estimated.
The two last parameters drive the differences between NS and Svensson. The latter is a
more flexible approach that allows a second “hump” in the forward rate curve and that provides
a better fit to the convex shape of the YC in the long end.
A number of authors have interpreted the first three parameters in the NS model, namely,
β0, β1, and β2, as the specific factors that drive the YC: level, slope, and curvature. β0 is related to
the long-run level of interest rates (the limit of the spot rate function as maturity approaches
infinity), β1 is regarded as the long-to-short-term spread (limit of the spot rate function as
maturity approaches zero minus β0), and β2 is related to the YC curvature, as its changes almost
exclusively affect medium-term interest rates. The literature focuses on the parameter 1. This
parameter determines the maturity at which the curvature factor β2 is maximized and the speed
of the exponential decay of the slope factor β1.
Diebold and Li (2006) introduce the dynamic NS model as a three-factor model to describe
YCs over time. To enable the estimation of time-varying latent factors in a linear setting, 1 is
kept constant over time. The authors argue that the loadings of the three factors are not very
sensitive to different values of 1 and thus fixed the parameter such that it maximizes the
loading on the curvature component at some medium term. This approach simplifies the
estimation of the remaining parameters to an ordinary least squares (OLS) regression with the
advantage of providing better numerical stability.26
We apply the Svensson model (the extended NS model) in its original static version in
order to obtain an accurate interest rate to price proposals. The numerical stability of the
The Svensson extension allows a second hump and s-shapes to be captured, but these shapes rarely appear in
term structures shorter than 20 years, which does not apply to our sample. Further, Bolder and Stréliski’s (1999)
comparison among NS, Svensson, and Super-Bell models is not conclusive. Regarding several goodness-of-fit
measures, they observe: “The Nelson-Siegel model appears to perform better than the Svensson model for the flat and
inverted term structures and worse for an upward-sloping yield curve. In aggregate, they even out and show little difference.”
In this sense, Diebold and Li (2006) enumerate a number of authors who have proposed extensions to NS to
enhance the flexibility (including Svensson) of the model; however, they conclude that from the perspective of
interest rate forecasting accuracy, the desirability of these generalizations is not obvious.
25 We do not analyze results for the estimated US zero-coupon YC by using unweighted and weighted versions of
Vasicek and Fong (1982) and NS (1987) models. We do not consider these results for the sake of brevity. These
results are available upon request from the authors, however.
26 Diebold and Li (2006) justify this method in terms of its simplicity, convenience, and “numerical trustworthiness by
enabling us to replace hundreds of potentially challenging numerical optimizations with trivial least-squares regressions.”
24
10
estimated parameters time series is of no concern to us. Thus, we estimate the six parameters
according to the original proposal of Svensson. Our experience shows that the parameters τ1
and τ2 play a relevant role, as these parameters determine inflection points and the position of
the possible two humps in the YC. The simultaneous estimation of all of the parameters
improves the fit quality but requires the use of nonlinear optimization techniques. In addition,
the values of both τ1 and τ2 are much more volatile than the values of the other four parameters.
The NS and Svensson methods have a number of advantages over spline methods: they are
simple, they result in more stable YCs, they require fewer data points, and they do not require
one to find an appropriate location for knot points, which joint up a series of splines. However,
spline methods allow for a much higher degree of flexibility than parametric models.
Specifically, the individual curve segments can move almost independently of each other
(subject to the continuity and differentiability constraints) so that separated regions of the curve
are less affected by movements in nearby areas. Therefore, spline approaches incorporate a
wider variety of YC shapes than the Svensson method.
2.2.4. The assumption about the error term variance
The three considered YC datasets use different fitting techniques. To illustrate the impact
of the fitting approach in the shape and stability of YCs, we apply the same method (the
Svensson method) to the same baskets of assets (samples A, B, C, and D) but introduce an
apparently insignificant modification in the error term variance assumption. We analyze the
impact of the structure of the variance of the error term εk of model [1] on the resulting
interest rate behavior.
The estimates of the term structure of interest rates usually assume homoscedasticity. In
the case of the Svensson method, the model can be estimated by OLS.
VAR[ 2k ]  2
[3]
However, this assumption is not neutral. In fact, a small error in a short-term bond price
produces an important error in its yield to maturity. By contrast, a large error on the price of a
long-term bond very slightly affects its yield to maturity. We should not forget that, the
dependent variable in this model is bond prices. Therefore, if we assume homoscedasticity, we
assign the same importance to errors in the price of all bonds, which means that we very heavily
penalize errors in the yields of long-term bonds. Therefore, assuming homoscedasticity implies
forcing the adjustment in the long end of the YC but at the cost of relaxing the adjustment of
the curve for short maturities.
To correct this problem, some authors suggested penalizing the valuation errors of the
short-term bonds; in particular, recommendations usually suggest correcting the variance of the
error term, making it proportional to the bond duration:
 P
VAR[ ]   k
 y k
2
k
2

D P
   2   k k

 1  yk
2

   2

[4]
where Dk is the k-bond duration, yk is its yield to maturity, and Pk is its price. Then, the model is
adjusted by using generalized least squares (GLS).
In this way, we force the adjustment of the short-term interest rates. However, this
adjustment is not free: it implies that we relax the adjustment of long-term interest rates.
3. Empirical analysis
3.1. The impact on the spot rates
11
As mentioned above, a direct comparison of the fitting accuracy between the considered
three external YC datasets is not possible, as differences in the prices/yields that are used as the
input and in the considered basket of securities prevent a direct comparison. First, we illustrate
the impact of the YC dataset provider’s fitting method decision by introducing a minor change
in the way that the same method is fitted from the same dataset. Second, we analyze the impact
of different datasets on the resulting YC despite the use of the same fitting method. Our
statistical analysis is accompanied by some examples to illustrate these problems. We examine
the YC estimates at three concrete dates. The first date (January 18th, 1996) is used to exemplify
differences in a simple aspect of the model choice, i.e., the error structure. For the second date
(July 5th, 2006), the shape of the observed yields to maturity of the securities included in the
sample is sufficiently complex to show differences in fitting flexibility. The third date
(September 9th, 1999) emphasizes the impact of the liquidity.
3.1.1. The fitting method
In this section, we show that even the same data sample and the same fitting method are
used, an apparently simple decision about the error term variance can imply large differences in
the behavior of the resulting spot rates. This exercise aims to illustrate the extent to which the
different fitting methods can affect the results of the analyses based on the three external YC
datasets considered in this paper.
Figure 1 depicts the term structure estimations for January 18th, 1996, for sample A (the full
sample). The solid line represents the Svensson model estimate using the heteroscedastic
structure (weighted Svensson, WSV). We can see that the curve describes the data for short
maturities quite well. By contrast, the dotted line represents the Svensson model estimate of the
term structure applying the unweighted or homoscedastic scheme for the variance of the error
terms (unweighted Svensson, USV). The adjustment is rather poor in the short end of the YC.
When making this assumption, we do not care about the fit for these short maturities in the
model; we care only about what happens on the opposite side.
[INSERT FIGURE 1]
We compute two measures to gauge the goodness of the fit for the daily estimates from the
two techniques. First, the root mean squared error (RMSE) shows the squared distance of the
theoretical prices from the observed prices. It provides a pseudo average error for the given set
of securities. Second, the mean absolute error (MAE) is the average distance between the
theoretical bill or bond prices and the observed ones in absolute value terms. This measure is
not as easily influenced by extreme observations as the RMSE measure.
Statistical results are summarized in Tables 3 and 4. Table 3 depicts the yearly average of
the estimated parameters of the Svensson model for the different samples (a function of the
considered baskets of securities) and for the different specifications of the error term variance
(unweighted, USV, and weighted, WSV, Svensson method) for the daily estimates (5728
individual dates). The yearly averages of the 0 parameter show the downward trend in the
long-term interest rates (level) during the 1996-2006 period. The negative values of the 1
parameter denote the predominant positive slope of the YC. As expected, excluding bills, onthe-run bonds, and first off-the-run bonds from the full sample slightly increases the level and
the slope of the YC. In general, the level of interest rates in the long end is, on average, higher
in the WSV than in the USV specification.
[INSERT TABLE 3]
Table 4 summarizes statistics for the goodness of fit of the estimated YC for the 5728
individual dates (1996-2006) using the USV method and the WSV method for the full sample
(sample A). Yearly averages of the RMSE and the MAE are classified in different bond maturity
tranches. As expected, the RMSE and MAE are lower in the WSV than in the USV for all
12
maturities, except for securities with maturities longer than 10 years. When the RMSE is
computed for all of the securities traded on each date, the global RMSE for the WSV is 2.3%
larger than that for the USV. Outstanding bonds with maturities longer than 10 years are less
numerous than those with shorter maturities, but their prices are more sensitive to interest
rates. As the RMSE squares the distance between observed and theoretical prices, a single large
error in a long maturity bond will have a larger relative contribution to the overall RMSE than
to the MAE. In fact, the global average MAE is 4.9% lower in the WSV estimates than in the
USV estimates. Differences in the short end are extreme. Thus, the average MAE in the WSV
estimates for securities with maturity equal to or less than 1 year is 63.2% lower than that in the
USV estimates.
[INSERT TABLE 4]
The results depend on the prevalent shape of the YC. Figure 2 plots the estimated YC from
the WSV method corresponding to the first working day of June during the eleven years of our
sample. For instance, the fit using the USV method for the short end of the YC is especially
poor during the year 2004. For 2004, the average RMSE and MAE for securities with maturity
equal to or less than 1 year are 80.0% and 84.8%, respectively, lower for the WSV method than
for the USV method. By contrast, the USV method clearly outperforms the WSV adjustment for
medium maturities (from 2 to 5 years) during the years 2003 and 2006. The average RMSE and
MAE for this maturity tranche are 32% and 38%, respectively, higher than for the WSV method
than for the USV method for these two years.
[INSERT FIGURE 2]
Based on these in-sample results, we may draw the conclusion that the WSV method fits the
bond data better than its unweighted counterpart. In the next section, we will only consider the
weighted version. Additionally, these preliminary results suggest that the assumption about the
structure of the variance of the error term, as an example of the difference in the fitting method,
affects not only the quality of the adjustment but also the results of further analyses based on
these YCs.
3.1.2. The data choice
In this section, we indirectly analyze the impact of using different baskets of securities to fit
YCs, which is one of the primary sources of discrepancies among the three considered external
YC datasets. A direct comparison of goodness of fit between these YC datasets is not possible
because they use different prices, security sets, and fitting methods. For this reason, we estimate
daily YCs by using the WSV method for baskets of securities that approximately mimic the
composition of those security datasets that are considered by the FRB (sample C, which
excludes all of the bills and the on-the-run and first off-the-run securities) and DoT (sample D,
which includes only the on-the-run bills, notes, and bonds for eleven maturities). We consider
our full sample (sample A) as the benchmark or basic sample. Sample A includes all of the
considered bills, notes, and bonds. Additionally, we analyze an intermediate case, sample B,
which includes traded bills and bonds that are neither on-the-run nor first off-the-run.
Our statistical analysis is preceded by two examples to illustrate the problem. We analyze
the YC estimates on July 5th, 2006. On this date, the shape of the observed yields to maturity of
the securities included in the sample is sufficiently complex to show differences in fitting
flexibility. The second date is September 9th, 1999, which clearly highlights the impact of
liquidity.
Figure 3 (panels A and B) depicts the original yields to maturity of the traded bonds, notes,
and bills in the US Treasury market on July 5th, 2006 (full sample or sample A) and alternative
YC estimates. On this date, yields to maturity show two humps. Finding a function that can
capture this double hump, which is observed quite often in the US market, can be difficult, as the
shape to fit this particular date is very complex. In the figure, the vertical axis represents
13
interest rates, and the horizontal axis represents the term to maturity of the securities traded in
that market. Further, the dots correspond to the observed yields to maturity of these securities,
and the lines represent estimations of the term structure using alternative models and datasets.
[INSERT FIGURE 3]
The upper panel (panel A) of Figure 3 allows a comparison of the three considered external
YC datasets, i.e., the YCs reported by the DoT, the FRB, and Bloomberg (F082), and our two
YC estimates (unweighted and weighted Svensson) for sample A of our bond price dataset
(GovPX). Mispriced zero-coupon interest rates can be observed in several maturity tranches. In
the very short end of the curve, the FRB curve does not fit the yields to maturity (the curve
provides very short-term interest rates, approximately 5.47%, while the observed ones are
approximately 4.85%). This result is surprising because the FRB uses the weighted version of
the Svensson method, which is particularly appropriate for these short maturities. The lack of
Treasury bills in the FRB sample composition most likely influences this result. In addition, the
DoT curve shows mispriced zero-coupon interest rates for maturities between 2 and 7 years.
Finally, a clear convexity problem appears in the long end of the term structure in the case
of the DoT, FRB, and F082 YCs on this particular date. Most flexible non-parametric methods
usually present a problem that prevents these long-term zero-coupon interest rates from
meeting the requirements to supply credible forward interest rates. The forward rates are very
sensitive to the shape of the YC, particularly in the very long end. The zero-coupon YC must be
asymptotically flat to provide a desirable flat forward rate curve.27
This problem should be addressed by using the parametric Svensson method used by the
FRB; however, the problem remains in this case.28 Gürkaynak et al. (2007) note that convexity
renders it difficult to fit the entire term structure of securities, especially securities with
maturities of twenty years or more. They maintain that convexity tends to pull down the yields
on longer-term securities, giving the YC a concave shape with longer maturities. This reported
problem in the FRB estimates generates three inconsistencies. First, long-term bonds cannot be
priced from these YCs. Their prices cannot be replicated from the zero-coupon interest rate
estimates. Second, for sufficiently long maturities, instantaneous forward rates become
negative.29 Third, the estimated value of the parameter β0 (long-run level of interest rates)
makes no economic sense for the 29% of the dates in the FRB YC dataset from 1996 to 2006 (it
is below 0.01%). In addition, this value is lower than 1% for 42% of the dates. These observed
inconsistencies of the FRB estimates do not appear in our USW and WSV estimations. We
obtain flat YCs for the longest maturities. A possible explanatory reason for this difference
could be the starting values of the six parameters of the Svensson model that are used in the
optimization. The final parameter vector of the model is known to be fairly sensitive to this
initial guess. Indeed, Table 5 (sample C and WSV) shows that our estimates converge to an
average estimated τ1 of approximately 1.78 years and a τ2 of approximately 2.04 years. For the
same sample period, the FRB reports average values of 1.46 and 13.19 years, respectively. These
parameters determine the location of the two possible humps.
The lower panel (panel B) of Figure 3 depicts four YC estimates from using the same fitting
technique applied to different baskets of securities. Now, we proceed to isolate the impact of the
asset sample composition on the resulting YCs. On this specific date, the YCs for samples A and
B move in parallel and show small differences between them. The YC for sample B fits the 1An investor’s expectation of the one-year interest rate likely to prevail in, for instance, 30 years should be the
same as the expectation of rates in 31 years.
28 The resulting instantaneous forward rates from the FRB are as follows: 5.64%, 10 years ahead, 5.00%, 20 years
ahead, 3.63%, 30 years ahead, and 3.50%, 31 years ahead.
29 Gürkaynak et al. (2006) recognize that the Svensson specification “assumes that forward rates eventually asymptote to
a constant. The downward tilt to forward rates at long horizons is an important characteristic of the U.S. yield curve; for
example, the instantaneous forward rate ending 25 years ahead has continuously been below the instantaneous forward rate
ending 20 years ahead for the past decade.”
27
14
year maturity better than the YC for sample A. The 1-year on-the-run bill included in sample A
pulls down the spot rates around this maturity. Moreover, the lack of the on-the-run and first
off-the-run bonds apparently has a small impact on the remaining maturities. The YC for
sample C, which also excludes the bills (mimicking FRB), also fit the 1-year maturity well, but it
does not fit the shortest maturities. In contrast, the fitting for maturities in the 15-year range
looks very good. The convexity problem observed in the FRB YC does not appear in our
estimate, which mimics its bond set. Finally, the YC for sample D is clearly conditioned by the
sample composition. The maturity of six of the eleven considered on-the-run securities is
shorter than 3 years. Consequently, this YC shows a good fit for short maturities. Meanwhile,
for maturities longer than 10 years, the resulting zero-coupon yields differ considerably from
the actual yields.
The impact of liquidity on the yields to maturity is clear on September 9th, 1999, as seen in
Figure 4. There is a wide gap in terms of the yield to maturity between the on-the-run bonds
and the off-the-run bonds.30 The model choice is not as relevant for this date, but the data
choice is crucial. Depending on this decision, we are, in fact, estimating different interest rates:
the spot rates corresponding to the average market liquidity level, the spot rates of the most
liquid references, or the spot rates of the seasoned bonds. The level of these interest rates should
be different, but most likely, the volatility and the correlation among forward rates are also
different.
[INSERT FIGURE 4]
The upper panel of Figure 4 illustrates differences in model choice and data choice between
the three external YC datasets. The two primary deficiencies observed in the upper panel of
Figure 3 (i.e., the poor fit of the FRB YC dataset for maturities shorter than 1 year and the
convexity problem in the long end of the three YCs) appear again in Figure 4. A new aspect is
clearly noted for this date: the DoT and F082 YCs are far from being smooth curves. Rather,
both methods show an over-fitting problem for this date. This saw-tooth pattern generates
serious inconsistencies in the forward rate term structure. Figure 5 plots the implied
instantaneous forward rates from the different YC datasets for this date. The obtained
extremely sinuous forward curves from the DoT and F082 data are clearly nonsense. However,
even in the FRB case, the downward sloping curve for maturities longer than 10 years also
makes no economic sense.
[INSERT FIGURE 5]
The lower panel of Figure 4 reveals the implications of the data choice, as it shows four YCs
for the same fitting technique (WSV) but with different security baskets. We can outline at least
four points. Including on-the-run securities only appears to pull the YC from sample A slightly
down for maturities longer than 10 years. The lack of bills in sample C (mimicking FRB) again
implies a bad fit in the short end of the YC. As sample D (mimicking DoT) includes only onthe-run bills and bonds, the fit is good for the short end of the YC, but it provides low spot rates
for most maturities.
Table 5 presents results for three of the proposed baskets of securities (samples B, C, and D)
on the quality of the fit from the WSV technique. Panel B of Table 4 depicts the full sample
(sample A) case. The RMSE and MAE are computed exclusively from the securities that are
included in each sample. This fact is a clear disadvantage for sample A. As Table 2 shows,
sample A includes 132 securities per date on average; meanwhile, sample C considers 96
securities per date, excluding bills and the frequently difficult to price on-the-run and first offthe-run bonds.
[INSERT TABLE 5]
The period from fall 1998 until the end of 1999 is characterized by several episodes of flights to quality. The
Year 2000 bug (Y2K) effect is suggested to be one of the triggers.
30
15
The only difference between sample B and the full sample (sample A) is the lack of on-therun and first off-the-run securities. According to the average RMSE and MAE, sample B has
better goodness of fit of than sample A in almost all maturity tranches and years. Indeed, the
average RMSE (MAE) for sample B is 24.6% (15.4%) lower than that for sample A. Removing
the on-the-run securities improves the performance of the fitting model for the remaining
securities. These issues often trade at a premium over other Treasury securities, owing to their
greater liquidity and their frequent specialness in the repo market. Thus, the liquidity of the
remaining securities is more homogenous.
Sample C mimics the FRB sample and thus considers neither bills not on-the-run and first
off-the-run bonds. As suggested by Figures 3 and 4, the fitting performance in the short end of
the YC is worse for sample C than for sample B. For bonds with remaining maturities equal to
or shorter than 1 year, the average RMSE and MAE are 33.5% and 93.8%, respectively, worse
for sample C than for sample A. For the full maturity spectrum, sample C improves the average
RMSE by 17.6% with respect to sample A, but in terms of average MAE, this improvement falls
to an irrelevant 1.3%. Gürkaynak et al. (2007) justify the decision to exclude bills since they are
often disconnected from the rest of the YC because of the segmented demand of money market
funds. If we only consider bonds with a remaining maturity longer than 1 year, the
improvement accounts for 30% in terms of both RMSE and MAE.
Sample D mimics the DoT sample and thus considers only eleven maturities. Our fitting
results from this sample are very disappointing. The WSV estimation from the prices of only
eleven on-the-run securities is very poor, especially for the short end and the long end of the YC
(maturities shorter than 1 year or longer than 10 years). On average, the RMSE and MAE are
92% higher for this sample than for sample A. This result suggests that this sample would not
be appropriate for pricing proposals.
An immediate implication of these results should appear in the correlations among the pairs
of forward rates. Forward rates play a key role in many financial issues, such as in the
implementation of interest rate models (e.g., Heath, Jarrow, and Morton), in many product
valuations where correlations among forward rates are crucial (e.g., swaptions), or in analyses of
market expectations for future short-term interest rates. A desirable property of any YC is that
it should produce an asymptotically flat forward rate curve. Forward rates are highly sensitive
to the shape of the YC, particularly in the very long end.
We compute correlations by using a 30-day window between pairs of several year ahead
forward rates with a six-month tenor.31 The differences between YC datasets for long
maturities are huge. For instance, we observe that the correlation between the 10-year forward
rate and the 30-year forward rate from the FRB and F082 datasets is close to zero
(uncorrelated) during most of the sample period. The results for sample A using the USV
method produce an almost perfect, positive correlation. Further, this method best fits the long
end of the YC.
At first glance, these divergences in the estimated YC affect a number of aspects, such as the
level, shape, temporal dynamic, forward curve, and so forth. The implications for the multiple
proposals that use this YC as the input are obvious. We speculate that all these elements may
have a statistically and economically significant impact on the results of various academic and
professional financial work for which these YCs can be used as the input.
3.2. The impact on the term structure of volatilities
In the previous section, we observed that different sample compositions and fitting
techniques can lead to differences in the level of interest rates for different maturities and
different fitting qualities for the overall YC. From this result, the time series behavior of the
interest rates is presumed to depend on the YC dataset. In this sense, Díaz et al. (2011) observe
31
These results are available upon request from the authors.
16
a significant impact from the chosen model when the zero-coupon YC is fit on the volatility
estimates.32
We perform an analysis of the VTS. From the three external YC datasets and our YC
estimates using the weighted Svensson technique from the four baskets of securities (samples A,
B, C, and D), we extract spot rates for 14 maturities ranging from one month up to 30 years.
These rates represent the input for two alternative methods to estimate the VTS. First, we
calculate simple standard deviation measures by using 30-day rolling windows from the log
difference of the value of the spot rates. We call the resulting annualized volatilities “historical
volatilities.” Second, we considered different specifications for the well-known family of the
conditional volatility models. According to the Schwarz and Akaike Information Criterion (SIC
and AIC, respectively), we choose the EGARCH(1,1) model proposed by Nelson (1991), which
allows for asymmetric impacts for the innovations.33
Table 6 summarizes various statistical results from the VTS estimations. In the case of
historical volatility, the results for the shortest maturities show a higher average level and
standard deviation of the VTS for the FRB dataset. The volatility of the VTS for the 1-month
maturity is almost explosive. This result reflects the fitting problem of this method in the very
short end of the curve. For the rest of the maturity spectrum, the VTS of the F082 is slightly
higher and more volatile. The over-fitting of this method and the sample composition may
explain this result. Regarding the computation of the EGARCH volatilities, the results are less
conclusive.
[INSERT TABLE 6]
The shape of the VTS changes depending on the weighting scheme that we chose to
estimate the term structure of interest rates. From the same data sample (sample A), the USV
method provides a higher (lower) volatility level for the shortest (largest) maturities than the
WSV method. The weights in the WSV provide more accuracy and stability in the short end of
the curve but are less accurate and stable in the long end.
If we consider our full sample (sample A) as the benchmark, we observe that the three
external datasets show a lower level of volatility in the short end of the YC (maturities shorter
than 1 year) than those shown by sample A. However, the F082 YC shows higher volatility for
maturities longer than 5 years. Excluding on-the-run securities (sample B) reduces the volatility
for the entire maturity spectrum. The same results are also observed when the bills are also
excluded (sample C). The Svensson technique applied to the 11 maturities used by the DoT
(sample D) shows the highest volatility levels among the eight considered samples. The
remaining results depend on how volatility is computed.
To test whether the observed differences in the VTS are significant from a statistical point
of view, we apply a sign test. This test allows us to check whether two alternative models
produce significant differences in the resulting spot rate volatilities.34 We observe significantly
different VTSs for most of the pairs of datasets—differences that appear to affect all maturities.
Díaz et al. (2011) apply NS’s (1987) and Vasicek and Fong’s (1982) methodologies to the Spanish Treasury debt
market.
33 Andersen and Benzoni (2007) have documented that an EGARCH representation for the conditional yield
volatility provides a convenient and successful parsimonious model for the conditional heteroskedasticity in these
series.
34 As the null hypothesis, this test assumes that given two alternative models to estimate zero-coupon bond yields,
the probability that one of them produces a higher volatility estimate than the other for a given day is 50%. Because
we use a 30-day window to estimate the volatilities, we select one of thirty estimates from our 2717 daily volatility
estimates to avoid autocorrelation problems. Eventually, we have 91 independent volatility estimates for each
maturity. Under the null hypothesis, the number of times that one method produces a higher volatility estimate
than the other (x) is distributed according to a binomial random variable with parameters N=91 and p=0.5. The
results are available upon request from the authors.
32
17
In particular, the most rigid model appears to provide the least volatile zero-coupon rates for
the shortest and longest maturities.
3.3. The impact on the expectations hypothesis
The expectations hypothesis has received considerable attention in the empirical literature.
Although empirical researchers have frequently rejected the expectations hypothesis, the
empirical evidence varies from one study to the next depending on the precise implication
tested, the segment of the YC examined, or the period studied. Most studies find little empirical
support for the theory, noting that the expected return premium is not constant. Nevertheless,
the literature is still not unanimous about the theory’s plausibility with respect to very shortterm or to long-term rates. Campbell and Shiller (1991) reject the expectations hypothesis for
all combinations of the short- and long-term rates when the long-term maturity is less than
four years but only reject it in one case when the long-term maturity is four years or greater.
Brooks et al. (2012) conclude that forward rates have the power to forecast spot rates but not
return premiums. Focusing on the very short end of the repo term structure, Longstaff (2000)
finds support for the expectations hypothesis. By contrast, Downing and Oliner (2007) and
Brown et al. (2008) find that calendar-time-based regularities lead to the rejection of the
expectations hypothesis at the very short end of the commercial paper term structure.
In this section, we analyze the impact of the different YC datasets on the expectations
hypothesis tests. One criterion to propose a YC dataset as the best dataset could be to explore
which interest rates are closer to supporting this theory. This study does not claim to provide a
comprehensive and detailed analysis of the expectations hypothesis but rather aims to highlight
the implications of using divergent interest rates. We do not perform a rigorous econometric
analysis. We merely replicate the classical Campbell (1995) test for our YC datasets. As a slight
robustness improvement, we use an instrumental variables regression. Campbell and Shiller
(1991) suggest this technique because, otherwise, the regression results should be extremely
sensitive to measurement error in the long-term interest rate.
We explore the expectations hypothesis by using the same model, the same interest rate
maturities, and the same sample period, but we use three alternative external datasets and our
own estimates. We do not aim to find evidence supporting or rejecting the theory; rather, we
rank the considered YC datasets as a function of their ability to approach the theory.
Following the detailed description of Campbell (1995), we compute continuously
compounded yields by using one month as the basic time unit. The yield spread of a bond is
defined as the difference between its yield and the short yield. As the short yield, Campbell
(1995) uses the yield of a 1-month Treasury bill. As the DoT reports yields of on-the-run
securities for several fixed maturities, our actual 1-month yield is proxied by the 1-month yield
reported by the DoT dataset.35 The excess return is calculated as the difference between the
return on a bond and the short yield. The excess return is also the yield spread less (m - 1) times
the change in the bond yield, where m is the maturity in months.
The expectations hypothesis implies that excess returns on long bonds over short bonds are
unforecastable, with a zero mean in the case of the pure expectations hypothesis. The first row
of Table 1 in Campbell (1995, page 135) checks whether the excess return on long bonds over
short bonds has a zero mean.36 We replicate this table for our analyzed YC datasets and even for
The DoT provides market yields at fixed maturities calculated from composites of quotations obtained by the
Federal Reserve Bank of New York. According to the information on the DoT website, the yields are investment
yields or bond equivalent yields. The formula uses simple interest and day count convention actual/actual. We
recalculate this yield as a continuosly compound interest rate. No 1-month rates are available in this dataset until
August 1, 2001. Prior to this date, we estimate the corresponding interest rate by using cubic interpolation.
36 The data in Table 7 are reported in annualized percentage points, i.e., the natural monthly variables are
multiplied by 1,200.
35
18
Campbell’s results in Table 7. The excess returns in Campbell’s original results show a hump
with a maximum at a 1-year maturity. Instead of a hump, we observe that excess returns rise
with the maturity. Further, the average values of excess returns for the shortest maturities
remain close to zero, except in the case of FRB. Thus, we cannot reject the null hypothesis of a
zero mean for all maturities shorter than 2 years.
[INSERT TABLE 7]
Campbell (1995) also analyzes whether excess returns on long bond over short bonds vary
predictably over time. Under the expectations hypothesis, the excess returns should be
unforecastable over the life of the short bond. Our Table 8 replicates Campbell’s Table 2. The
first row of Table 2 reports the slope coefficients from a series of regressions of long rate
changes on a constant and the long-short yield spread. If the expectations hypothesis holds, the
slope coefficient should be one. In Campbell’s original table, all but the first of the slope
coefficients are negative, and they decline with maturity. Our slope coefficients are positive for
maturities until 1 year (2 years in our estimated YC from GovPx). Moreover, they are even
close to the desirable value of one for the shortest maturities in three cases: the DoT, sample A,
and sample B YC datasets. Another notable result is that the estimated slope coefficients have
similar values in all of the considered YC samples for maturities of 1 year or longer. Thus,
differences in model choice and in data choice for computing YCs appear to affect this simple
expectations hypothesis test in only the short end of the curve.
[INSERT TABLE 8]
Finally, if the expectations hypothesis holds, then the excess return on a long bond over a
sequence of short bonds must also be unforecastable over the life of the long bond. In this case,
the slope coefficient for the regressions of long-run changes in short-term interest rates onto a
constant and the yield spread would be one. These slope coefficients are reported in the second
row of Campbell’s original Table 2 and in the second row of each of the panels of our Table 8.
Campbell finds some support for the expectations hypothesis over horizons of two or three
months and horizons of several years. In our case, we observe slope coefficients close to one (or
even considerably higher in the DoT, sample A, and sample B YC datasets) for all maturities
until 2 years. Thus, high yield spreads are associated with rising short-term interest rates,
except for 10-year maturities. In this case, the slope coefficients are almost identical in all of the
considered YC samples for maturities longer than 1 year.
The results of the three analyses that we conduct provide some evidence that supports the
expectations hypothesis in the short end of the YC. In general, for maturities equal to or shorter
than 1 year, we observe that the excess returns on long bonds over short bonds have a zero
mean. In addition, the yield spreads appear to forecast short-run changes in long yields and
long-run changes in short yields. These results can be hampered by the very simple techniques
that we use. Nevertheless, they show some divergence among our YC datasets. The lack of bills
in the FRB sample and the mixture of assets in the F082 sample could generate a poor fit in the
short end of the YC. The results for longer maturities are quite similar among the different YC
datasets. For these maturities, model choice and data choice appear to have no impact on the
outcomes of the tests for the theoretical behavior of interest rates.
4.- Guidance for final users and conclusions
Alternative YC datasets are usually assumed to be perfectly accurate when they are used as
the input for a number of financial purposes. Practitioners and researchers thus often do not
question the accuracy of YCs that they obtain from data providers. In this paper, we examine
three popular datasets and our own estimates of the zero-coupon bond yields from the Treasury
market. Specifically, we analyze the differences in terms of the fitting methodology, the basket
of securities that are included in the sample, and the specific prices or yields that are used as the
19
input. From our analysis, we extract a number of suggestions to help financial data users select
a YC dataset.
The US DoT provides YCs that are computed from yields to maturity of only 11
government securities (bills and bonds). These securities are the on-the-run for each maturity
(except for the 20-year composite rate, which is included by completing the maturity spectrum).
The YC is estimated from the most liquid assets in the market and with a clear bias to focus on
short maturities. We observe a very good fit for short maturities. Further, the YC shape is quite
simple but unstable for intermediate and long maturities, as more than half of the sample are
assets less than 3 years. This dataset is appropriate for when spot interest rates for the most
liquid short-term bills and bonds in the market are needed and when theoretical tests that
involve short-term maturities are used.
The FRB YC dataset only considers straight second-off-the-run or older bonds. Thus, bills
and the most liquid bonds (on-the-run and first-off-the-run) are excluded. The FRB dataset
poorly fits short maturities, which also generates a stability problem in the short end of the
curve. This deficiency has critical implications for interest rate option models, models that are
calibrated to estimated YCs, and analyses of market expectations for future short-term interest
rates. The fit performance in intermediate and long maturities is good, and the extra-good fit
for periods of flights to liquidity is especially remarkable. In addition, the YC shows a low level
of volatility for the entire maturity spectrum except for the shortest maturity. Financial
academics and practitioners who require good interest rate estimates for a regular liquidity level
in a maturity range between 1 and 10 years can trust the FRB dataset.
The F082 Bloomberg YC dataset presents an over-fitting problem. Bloomberg considers all
of the outstanding government securities as inputs, including traded securities, non-traded
securities, and callable old bonds. The methods for calculating the “generic” prices and fitting
the YC are not publicly available. The extremely sinuous shape of the YCs demonstrates the
great flexibility of the technique, which is appropriate because it almost exactly estimates the
prices of all of the securities considered by Bloomberg. The YCs show frequent saw-tooth
patterns, which can generate serious inconsistencies in the forward rate term structure. Further,
these YCs depend on idiosyncratic factors and bond-specific factors that are irrelevant to
pricing out-of-sample securities. Thus, Bloomberg suffers from instability in estimated YCs. In
fact, it offers the poorest results relative to other alternatives. We recommend using this YC
dataset to obtain a view of the current full market situation.
None of the three alternative YC datasets performs well for maturities longer than 10 years.
The convexity problem and the uncorrelated long-term forward rates are common problems.
The Svensson approach employed by the FRB should provide a desirable flat forward rate
curve, but it does not. Additionally, this YC dataset shows incongruent negative values for the
estimates of the β0 parameter (long-run level of interest rates) for 29% of the dates. Our YC
estimates from GovPx data using the same method and a sample that mimics the one used by
them do not reproduce these problems. A review of the parameters’ initial guess and the
optimization program would most likely improve the FRB results.
These observed differences have serious negative implications for academics and researchers
who assume that the YC that they use is irrelevant and who are not concerned about intrinsic
differences between alternative popular YC datasets. These differences also have negative
implications for practitioners who indiscriminately use these YC datasets. As our analysis
indicates, practitioners should pay more attention to the dataset that they use and should
determine which dataset is the most accurate for their concrete valuation purposes.
20
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present”, Journal of Monetary Economics, Vol. 54, no. 8, pp. 2291-2304
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21
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22
Figure 1.- The importance of the variance of the error term
Note: the points represent the yields to maturity of all of the non-callable traded Treasury securities (sample A or
the full sample).
23
Figure 2.- Term structure of interest rates (1996-2006)
First working day of June during the sample period. YCs fitted according to weighted version of
Svensson’s (1994) model from GovPx asset prices (sample A).
8%
1996
1997
7%
2000
1999
6%
2006
Interest rates
1998
5%
2001
2004
2002
2005
4%
3%
2003
2%
1%
0%
0
5
10
15
20
Term to maturity (years)
24
25
30
Figure 3.- Alternative estimations for the term structure of interest rates
Panel A. Different fitting methods and different baskets of assets
July 5, 2006
5.6
F082
5.5
WSV
FRB
USV
DoT
Interest rates (%)
5.4
DoT
F082
5.3
5.2
USV
FRB
Yield to maturity
Unweighted Svensson (USV)
5.1
Weighted Svensson (WSV)
Federal Reserve Board
5
Department of Treasury
Bloomberg (F082)
4.9
4.8
0
5
10
15
20
25
30
Term to maturity (years)
Panel B. Same fitting methods (weighted Svensson) and different baskets of assets (from the
GovPX dataset)
July 5, 2006
Sample C
Sample C
5.55
Sample B
Interest rates (%)
5.45
Sample D
5.35
5.25
Yield to maturity
5.15
Sample A
5.05
WSV (A: full sample)
WSV (B: excluding OTR)
Sample D
WSV (C: excluding bills & OTR)
WSV (D: only OTR)
4.95
4.85
0
5
10
15
20
Term to maturity (years)
25
30
Note: the points represent the yields to maturity of all of the non-callable traded Treasury securities (sample A or
the full sample).
25
Figure 4.- The impact of liquidity on yields to maturity
Panel A. Different fitting methods and different baskets of assets
Panel B. Same fitting methods (weighted Svensson) and different baskets of assets (from the
GovPX dataset)
Note: the points represent the yields to maturity of all of the non-callable traded Treasury securities (sample A or
the full sample).
26
Figure 5.- The impact on the instantaneous forward interest rates
DoT and F082 datasets provide interest rates for certain fixed maturities. We use a simple cubic
interpolation technique to build continuous YCs for these two datasets. For each maturity n, the
corresponding instantaneous forward interest rate is proxied by the 1-week rate beginning n years
ahead.
Panel A. Date analyzed in Figure 3.
Panel B. Date analyzed in Figure 4.
27
Table 1. Primary characteristics of the considered YC datasets
Gürkaynak, Sack &
Wright (2007) in
Federal Reserve Board
(FRB)
U.S. Department of the
Treasury (DoT)
Bloomberg (F082)
Bills
On-the-run
bonds and
notes
Remaining
straight bonds
and notes
Nonstraight
bonds
(callable…)
Shortest
maturity
End-of-day prices
No
No
Yes
(excluding the
first off-the-run)
No
3-month
Close of business bid
yields-to-maturity
Yes (only 4-,
13-, 26-, 52weeks)
Yes (only 2-,
3-, 5-, 7-, 10-,
20-, 30-years)
No
No
1-month
Bloomberg generic
prices (quotes over a
time window)
No
Yes
Yes
Yes
Not reported
- Bills: no
restrictions
- Bonds: 1
year (6
month from
2001*)
Fitting
technique
Market data
Weighted
Svensson
(1994)
Quasi-cubic
hermite
spline
function
Piecewise
linear
function
Sample A
(full sample)
Weighted
Svensson
(1994)
Last trade price if
available or mid-quote
at 5 pm from GovPX
Yes
Yes
Yes (only the
youngest bond if
same maturity)
No
Sample B
(no bills)
Idem
Idem
Yes
No
Idem
Idem
Sample C
(mimicking FRB)
Yes
(excluding the
first off-the-run)
Idem
Idem
No
No
Idem
Idem
Idem
Sample D
(mimicking DoT)
Idem
Idem
Yes (only 4-,
13-, 26-, 52week)
Yes (only 2-,
3-, 5-, 7-, 10-,
20-, 30-year)
No
Idem
Idem
Because the 1-year Treasury bill is no longer auctioned beginning March 2001, we also consider Treasury notes and bonds with remaining maturities between 6 and 12
months after 2001.
Note: “Idem” means same contain that the upper cell.
*
28
Table 2. Composition of our dataset from GovPx asset prices
This table shows the average number of observations per day, the number of trading days in the year
and the average maturity of the longest bond included in the daily estimation.
(A) Full sample: bills, notes, and bonds
(B) Full sample excluding on-the-run and first off-the-run
(C) Mimicking FRB (excluding bills, on-the-run, first off-the-run)
(D) Mimicking DoT (only on-the-run) (11 maturities)*
# observations per day
Days
Maturity
Year
(A)
(B)
(C)
(D)
per year
longest bond
1996
151.6
137.6
115.2
11.0
259
29.8
1997
150.1
136.2
114.0
11.0
261
29.8
1998
146.7
132.7
109.4
11.0
261
29.7
1999
134.4
120.4
96.8
11.0
261
29.8
2000
120.0
106.1
84.6
11.0
260
29.8
2001
114.6
100.6
79.6
11.0
258
29.5
2002
113.3
99.3
76.7
11.0
261
28.6
2003
115.2
101.2
78.6
11.0
261
27.6
2004
123.8
109.8
87.2
11.0
262
26.6
2005
129.6
115.7
93.7
11.0
260
25.6
2006
157.0
143.4
122.0
11.0
260
28.6
Avg
132.4
118.4
96.1
11.0
260
28.7
Note: all of the samples exclude “when-issued” and cash management transactions, trades and quotes related to callable bonds,
and TIPS, as well as outliers (usual filters are applied).
* Four maturities of the most recently auctioned bills (4-, 13-, 26-, and 52-weeks), six maturities of just-issued bonds and notes
(2-, 3-, 5-, 7-, 10-, and 30-years), and the composite rate in the 20-year maturity range.
29
Table 3. Estimated primary parameters of the Svensson (1994) model from GovPx asset prices
(A) Full sample (bills, notes, and bonds)
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
96/06
0
7.21
6.94
6.00
6.34
6.03
6.13
6.31
6.23
6.14
5.01
5.10
6.13
Unweighted Svensson (USV)
1
2
1
3
-0.27
-0.08
0.73
0.12
-0.21
-0.07
0.69
0.10
-0.54
0.14
0.96
0.26
-1.08
0.45
0.83
0.53
-0.19
0.11
0.74
0.09
-1.02
0.41
0.95
0.49
-0.08
-0.03
1.34
0.02
-0.10
-0.03
1.72
0.03
-0.08
-0.02
2.00
0.02
-0.84
0.36
1.30
0.41
-0.90
0.41
1.13
0.45
-0.48
0.15
1.13
0.23
2
1.48
1.51
1.47
0.95
0.91
1.00
1.31
1.72
2.00
1.35
1.17
1.35
0
7.27
6.98
6.08
6.49
6.10
6.20
6.36
6.11
6.21
5.03
5.09
6.18
Weighted Svensson (WSV)
1
2
1
3
-0.21
-0.04
1.06
0.09
-0.18
-0.03
0.97
0.08
-0.25
0.01
1.13
0.12
-0.72
0.31
1.51
0.35
-0.75
0.37
0.85
0.37
-0.71
0.30
1.48
0.34
-0.71
0.28
1.45
0.33
-0.69
0.30
2.46
0.32
-0.69
0.25
1.70
0.32
-0.57
0.21
1.18
0.28
-0.19
0.03
0.79
0.09
-0.52
0.18
1.33
0.25
2
1.96
2.09
1.76
1.58
0.88
1.49
1.45
2.36
1.80
1.35
1.17
1.63
0
7.28
6.98
6.09
6.50
6.12
6.21
6.40
6.34
6.24
5.06
5.11
6.21
Weighted Svensson (WSV)
1
2
1
3
-0.22
-0.05
1.00
0.10
-0.19
-0.04
0.92
0.08
-0.25
0.03
1.28
0.12
-0.73
0.32
1.43
0.35
-0.74
0.36
0.85
0.37
-0.76
0.33
1.45
0.37
-0.71
0.28
1.39
0.33
-0.42
0.12
1.81
0.18
-0.68
0.24
1.73
0.31
-0.47
0.15
1.18
0.23
-0.15
0.00
0.77
0.07
-0.48
0.16
1.25
0.23
2
1.79
1.68
1.78
1.50
0.88
1.46
1.41
1.86
1.82
1.40
1.22
1.53
0
7.32
7.03
6.14
6.46
5.98
6.25
6.46
6.36
6.27
5.13
5.20
6.24
Weighted Svensson (WSV)
1
2
1
3
-0.91
0.41
1.95
0.44
-0.91
0.42
1.94
0.45
-0.91
0.41
2.05
0.45
-1.01
0.46
1.42
0.49
-0.08
-0.02
2.16
0.06
-0.91
0.40
1.56
0.44
-0.92
0.38
1.52
0.44
-0.92
0.38
1.90
0.43
-0.93
0.38
1.82
0.44
-0.91
0.40
1.54
0.45
-1.02
0.48
1.70
0.51
-0.86
0.37
1.78
0.42
2
2.01
1.99
2.11
1.47
4.57
1.58
1.55
1.91
1.87
1.61
1.73
2.04
0
7.09
6.87
5.92
6.28
5.95
6.07
6.16
6.09
6.06
4.95
4.98
6.04
Weighted Svensson (WSV)
1
2
1
3
-0.21
-0.03
0.84
0.09
-0.18
-0.03
0.87
0.08
-0.26
0.02
1.04
0.12
-0.70
0.30
1.33
0.34
-0.73
0.37
0.78
0.36
-0.69
0.30
1.55
0.33
-0.71
0.29
1.43
0.33
-0.67
0.25
1.78
0.31
-0.67
0.24
1.75
0.31
-0.47
0.15
1.14
0.22
-0.13
-0.01
0.64
0.06
-0.49
0.17
1.20
0.23
2
1.48
1.59
1.48
1.40
0.80
1.56
1.43
1.79
1.84
1.36
1.09
1.44
(B) Full sample excluding on-the-run and first off-the-run
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
96/06
0
7.24
6.95
6.02
6.37
6.10
6.15
6.34
6.28
6.19
5.05
5.13
6.17
Unweighted Svensson (USV)
1
2
1
3
-0.26
-0.09
0.73
0.12
-0.21
-0.07
0.70
0.10
-0.76
0.32
1.33
0.37
-1.13
0.51
0.89
0.55
0.03
-0.19
0.36
0.10
-1.18
0.50
0.95
0.58
-1.16
0.44
0.96
0.55
-0.10
-0.03
1.80
0.02
-0.08
-0.02
2.07
0.02
-0.92
0.39
1.23
0.45
-0.88
0.40
1.21
0.44
-0.60
0.20
1.11
0.30
2
1.53
1.51
1.42
0.93
1.53
0.99
1.02
1.80
2.07
1.31
1.25
1.40
(C) Mimicking FRB (excluding bills, on-the-run, first off-the-run)
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
96/06
0
7.24
6.95
6.02
6.36
6.03
6.08
6.33
6.27
6.17
5.05
5.13
6.15
Unweighted Svensson (USV)
1
2
1
3
-0.27
-0.21
0.58
0.14
-0.20
-0.19
0.54
0.11
-0.22
-0.38
0.49
0.17
-1.17
0.52
0.85
0.57
-0.76
0.23
2.28
0.44
-1.05
0.42
1.06
0.50
-0.71
-0.18
0.61
0.45
-0.73
-0.33
0.61
0.35
-0.86
0.02
0.91
0.41
-1.03
0.44
1.21
0.50
-0.98
0.45
1.20
0.49
-0.72
0.07
0.94
0.38
2
1.44
1.43
1.36
0.89
6.24
1.11
1.23
1.27
1.35
1.28
1.24
1.71
(D) Mimicking DoT (only on-the-run) (11 maturities)
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
96/06
0
7.01
6.84
5.86
6.13
5.84
6.05
6.13
6.04
6.00
4.92
4.97
5.98
Unweighted Svensson (USV)
1
2
1
3
-0.27
-0.11
0.49
0.13
-0.20
-0.09
0.61
0.09
-0.96
0.42
1.22
0.47
-1.13
0.52
0.89
0.55
0.04
0.01
0.73
-0.02
-0.89
0.37
1.15
0.43
-1.09
0.43
1.10
0.52
-0.10
-0.03
1.72
0.02
-0.17
0.02
1.94
0.06
-0.91
0.39
1.21
0.45
-1.30
0.62
0.92
0.65
-0.63
0.23
1.09
0.31
2
1.12
1.42
1.29
0.91
0.96
1.19
1.15
1.72
1.95
1.28
1.05
1.28
30
Table 4.- Summary statistics of YC estimates using Svensson (1994): sample A
These tables report yearly averages of the root mean squared errors (RMSE) and mean absolute errors (MAE)
from the daily estimation (5728 individual dates) of the term structure of interest rates obtained by applying
unweighted (USV) and weighted (WSV) versions of Svensson (1994) to sample A from GovPx asset prices. Only
in-sample pricing errors are computed.
Panel A. Unweighted Svensson on the full sample (bills, notes, and bonds): sample A
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Avg
RMSE
0.304
0.255
0.437
0.717
0.523
0.602
0.738
0.600
0.467
0.340
0.286
0.479
MAE
0.150
0.138
0.243
0.389
0.291
0.354
0.429
0.326
0.277
0.202
0.154
0.268
Avg. RMSE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.021 0.051 0.079
0.290
0.658
0.022 0.053 0.069
0.284
0.506
0.048 0.086 0.128
0.541
0.801
0.073 0.057 0.174
0.871
1.246
0.066 0.108 0.183
0.458
0.870
0.045 0.077 0.194
0.864
0.901
0.060 0.078 0.207
1.137
1.098
0.043 0.047 0.120
0.944
0.889
0.079 0.069 0.109
0.742
0.699
0.042 0.066 0.076
0.466
0.566
0.022 0.045 0.062
0.324
0.547
0.047 0.067 0.127
0.629
0.798
Avg. MAE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.017 0.041 0.065
0.206
0.497
0.017 0.043 0.055
0.195
0.412
0.044 0.071 0.105
0.372
0.636
0.057 0.046 0.107
0.610
1.008
0.054 0.100 0.140
0.331
0.632
0.037 0.065 0.147
0.703
0.712
0.049 0.068 0.167
0.979
0.850
0.039 0.040 0.098
0.736
0.660
0.074 0.061 0.099
0.610
0.530
0.038 0.059 0.067
0.407
0.445
0.018 0.035 0.051
0.284
0.414
0.040 0.057 0.100
0.494
0.618
Panel B. Weighted Svensson on the full sample (bills, notes, and bonds): sample A
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Avg
RMSE
0.310
0.260
0.453
0.757
0.554
0.619
0.750
0.569
0.470
0.352
0.295
0.490
MAE
0.142
0.132
0.230
0.377
0.270
0.345
0.415
0.295
0.246
0.193
0.163
0.255
Avg. RMSE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.017 0.045 0.074
0.250
0.686
0.022 0.047 0.066
0.249
0.530
0.021 0.068 0.118
0.452
0.866
0.047 0.061 0.144
0.677
1.392
0.036 0.052 0.108
0.429
0.950
0.028 0.066 0.192
0.754
0.966
0.025 0.079 0.198
1.029
1.152
0.013 0.046 0.158
0.844
0.856
0.016 0.044 0.086
0.629
0.751
0.018 0.039 0.087
0.442
0.596
0.014 0.044 0.082
0.364
0.544
0.023 0.054 0.119
0.557
0.845
31
Avg. MAE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.012 0.036 0.058
0.174
0.499
0.014 0.039 0.051
0.162
0.417
0.015 0.054 0.092
0.289
0.677
0.029 0.054 0.085
0.402
1.106
0.018 0.044 0.084
0.301
0.667
0.018 0.052 0.155
0.583
0.749
0.015 0.065 0.165
0.846
0.880
0.008 0.040 0.136
0.628
0.614
0.011 0.038 0.074
0.500
0.549
0.013 0.034 0.077
0.382
0.454
0.010 0.035 0.070
0.325
0.411
0.015 0.045 0.095
0.417
0.639
Table 5.- Summary statistics of YC estimates using weighted Svensson (1994): samples B,
C, and D
These tables report yearly averages of the root mean squared errors (RMSE) and mean absolute errors (MAE)
from the daily estimation (5728 individual dates) of the term structure of interest rates obtained by applying the
weighted Svensson (WSV) to different datasets from GovPx asset prices. Only in-sample pricing errors are
computed.
Panel A. Weighted Svensson on the full sample excluding on-the-run and first off-the-run
securities: sample B
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Avg
RMSE
0.204
0.191
0.320
0.561
0.380
0.461
0.566
0.483
0.369
0.280
0.246
0.369
MAE
0.109
0.108
0.186
0.309
0.220
0.288
0.349
0.274
0.212
0.169
0.151
0.216
Avg. RMSE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.015 0.042 0.074
0.191
0.429
0.019 0.043 0.066
0.172
0.382
0.018 0.055 0.097
0.280
0.611
0.040 0.053 0.116
0.412
1.022
0.015 0.043 0.094
0.279
0.630
0.024 0.058 0.147
0.550
0.698
0.023 0.069 0.164
0.801
0.833
0.011 0.039 0.134
0.756
0.702
0.015 0.045 0.086
0.551
0.554
0.017 0.041 0.087
0.384
0.447
0.015 0.044 0.081
0.359
0.424
0.019 0.048 0.104
0.431
0.612
Avg. MAE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.010 0.033 0.059
0.148
0.328
0.013 0.035 0.052
0.138
0.306
0.013 0.046 0.078
0.217
0.525
0.024 0.046 0.072
0.275
0.883
0.011 0.036 0.077
0.228
0.515
0.016 0.046 0.125
0.482
0.587
0.014 0.057 0.144
0.709
0.700
0.008 0.033 0.117
0.574
0.563
0.010 0.039 0.074
0.430
0.443
0.012 0.036 0.077
0.334
0.368
0.011 0.036 0.069
0.324
0.356
0.013 0.040 0.086
0.351
0.507
Panel B. Weighted Svensson on the sample mimicking FRB (excluding bills, on-the-run, first
off-the-run): sample C
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Avg
RMSE
0.228
0.219
0.363
0.612
0.260
0.536
0.661
0.552
0.421
0.316
0.270
0.404
MAE
0.126
0.123
0.221
0.367
0.170
0.362
0.445
0.350
0.264
0.190
0.153
0.252
Avg. RMSE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.024 0.034 0.069
0.176
0.450
NA
0.024 0.055
0.145
0.415
NA
0.036 0.088
0.262
0.642
0.017 0.024 0.117
0.430
0.994
0.110 0.041 0.079
0.265
0.355
0.029 0.046 0.131
0.450
0.749
0.030 0.048 0.127
0.633
0.904
0.018 0.034 0.120
0.733
0.719
0.021 0.037 0.069
0.525
0.577
0.017 0.020 0.050
0.297
0.493
0.016 0.028 0.052
0.230
0.493
0.031 0.034 0.087
0.377
0.617
Avg. MAE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.024 0.025 0.052
0.137
0.344
NA
0.019 0.042
0.120
0.331
NA
0.028 0.070
0.203
0.551
0.017 0.019 0.070
0.284
0.859
0.110 0.034 0.058
0.193
0.286
0.024 0.036 0.109
0.380
0.628
0.025 0.040 0.107
0.528
0.756
0.015 0.029 0.104
0.552
0.575
0.017 0.031 0.059
0.405
0.459
0.014 0.016 0.041
0.250
0.394
0.013 0.019 0.041
0.194
0.398
0.029 0.027 0.068
0.295
0.508
Panel C. Weighted Svensson on the sample mimicking DoT (only on-the-run) (11 maturities):
sample D
Year
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
Avg
RMSE
0.584
0.479
0.850
1.307
0.995
0.967
1.162
0.927
0.700
1.798
0.474
0.931
MAE
0.319
0.279
0.496
0.753
0.546
0.561
0.660
0.502
0.373
0.684
0.260
0.494
Avg. RMSE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.021 0.037 0.068
0.347
1.265
0.023 0.034 0.067
0.412
0.971
0.039 0.075 0.157
0.743
1.725
0.041 0.080 0.248
1.020
2.766
0.068 0.082 0.156
0.588
2.182
0.034 0.063 0.246
0.811
2.035
0.022 0.101 0.225
0.891
2.510
0.011 0.031 0.086
0.730
2.010
0.016 0.030 0.060
0.478
1.557
2.303 0.038 0.098
0.344
1.178
0.011 0.029 0.073
0.304
1.035
0.235 0.054 0.135
0.606
1.749
32
Avg. MAE according bond maturity range
[0,1] [1,2] [2,5] [5,10] [10,30]
0.018 0.036 0.060
0.285
1.241
0.019 0.033 0.058
0.342
0.945
0.032 0.074 0.142
0.609
1.682
0.032 0.080 0.208
0.854
2.734
0.045 0.082 0.138
0.527
2.090
0.025 0.061 0.224
0.652
2.017
0.015 0.098 0.214
0.761
2.472
0.008 0.030 0.078
0.582
2.004
0.012 0.029 0.053
0.374
1.552
1.259 0.036 0.085
0.286
1.165
0.009 0.028 0.066
0.256
1.019
0.134 0.053 0.121
0.503
1.720
Table 6.- Summary statistics of VTS estimates
These tables report daily averages and standard deviations of the volatility term structure (VTS) computed by two
alternative methods: annualized simple standard deviation measures using 30-day rolling windows from the logdifference of the value of the spot rates (“historical volatilities”) and an EGARCH(1,1) model. As the input, we use
zero-coupon interest rates from the three external YC datasets (Department of the Treasury, DoT; Federal
Reserve Board, FRB; Bloomberg, F082) and from our YC estimates, applying the unweighted Svensson (USV)
method to sample A and the weighted Svensson (WSV) to the four different datasets from GovPx asset prices
(sample A: full sample (bills, notes, and bonds); sample B: full sample excluding on-the-run and first off-the-run;
sample C: mimicking FRB, excluding bills, on-the-run, first off-the-run; sample D: mimicking DoT, only on-therun, 11 maturities.
Panel A: Historical Volatility from standard deviations
t
1m
3m
6m
1 yr
1.5y
2 yr
3 yr
5 yr
7 yr
10y
15y
20y
25y
30y
DoT
n/a
0.19
0.18
0.22
n/a
0.27
0.25
0.22
0.20
0.18
n/a
0.15
n/a
0.13
FRB
0.58
0.24
0.18
0.24
0.26
0.26
0.25
0.22
0.19
0.17
0.15
0.14
0.13
0.14
F082
n/a
0.22
0.21
0.28
n/a
0.28
0.26
0.23
n/a
0.20
0.18
0.16
n/a
0.16
Mean
USV
A
0.57 0.55
0.39 0.36
0.28 0.24
0.25 0.25
0.27 0.27
0.27 0.28
0.27 0.27
0.23 0.23
0.20 0.20
0.17 0.18
0.15 0.16
0.14 0.15
0.14 0.15
0.14 0.16
B
0.57
0.37
0.24
0.24
0.27
0.27
0.26
0.22
0.20
0.17
0.15
0.14
0.14
0.14
C
0.34
0.26
0.23
0.25
0.27
0.27
0.26
0.22
0.20
0.17
0.15
0.14
0.14
0.14
D
0.81
0.44
0.27
0.28
0.29
0.28
0.27
0.24
0.21
0.18
0.16
0.16
0.17
0.18
DoT
n/a
0.12
0.12
0.16
n/a
0.19
0.16
0.11
0.08
0.06
n/a
0.04
n/a
0.04
FRB
1.70
0.20
0.13
0.19
0.20
0.19
0.15
0.10
0.07
0.05
0.04
0.03
0.03
0.04
Standard deviation
F082 USV
A
B
n/a
0.55 0.17 0.15
0.14
0.33 0.12 0.11
0.14
0.21 0.14 0.14
0.23
0.19 0.19 0.18
n/a
0.20 0.20 0.20
0.20
0.20 0.19 0.19
0.16
0.17 0.16 0.16
0.11
0.11 0.11 0.11
n/a
0.08 0.08 0.08
0.07
0.06 0.06 0.06
0.05
0.04 0.04 0.04
0.04
0.03 0.03 0.03
n/a
0.03 0.04 0.03
0.05
0.03 0.05 0.03
C
0.31
0.21
0.17
0.19
0.20
0.20
0.16
0.11
0.08
0.06
0.04
0.03
0.03
0.04
D
0.18
0.13
0.19
0.21
0.19
0.18
0.15
0.11
0.08
0.06
0.04
0.04
0.05
0.07
FRB
0.90
0.16
0.13
0.20
0.21
0.19
0.16
0.10
0.08
0.06
0.04
0.03
0.03
0.04
Standard deviation
F082 USV
A
B
n/a
0.28 0.15 0.23
0.12
0.18 0.10 0.16
0.12
0.14 0.11 0.13
0.20
0.17 0.18 0.18
n/a
0.19 0.20 0.19
0.19
0.19 0.19 0.18
0.15
0.16 0.16 0.16
0.11
0.11 0.12 0.11
n/a
0.08 0.08 0.08
0.06
0.06 0.06 0.06
0.05
0.04 0.04 0.04
0.04
0.03 0.04 0.03
n/a
0.03 0.04 0.03
0.04
0.03 0.05 0.03
C
0.27
0.19
0.16
0.19
0.20
0.19
0.16
0.11
0.08
0.06
0.04
0.03
0.03
0.03
D
0.15
0.09
0.13
0.18
0.17
0.16
0.14
0.10
0.08
0.06
0.04
0.04
0.04
0.05
Panel B. Conditional volatility from EGARCH(1,1)
t
1m
3m
6m
1 yr
1.5y
2 yr
3 yr
5 yr
7 yr
10y
15y
20y
25y
30y
DoT
n/a
0.19
0.18
0.23
n/a
0.28
0.27
0.23
0.21
0.19
n/a
0.15
n/a
0.14
FRB
0.40
0.19
0.18
0.25
0.27
0.27
0.26
0.23
0.20
0.18
0.16
0.14
0.13
0.14
F082
n/a
0.19
0.18
0.25
n/a
0.27
0.25
0.23
n/a
0.18
0.17
0.15
n/a
0.15
Mean
USV
A
0.31 0.50
0.23 0.30
0.20 0.19
0.24 0.24
0.26 0.27
0.27 0.27
0.26 0.26
0.23 0.23
0.20 0.20
0.17 0.17
0.15 0.15
0.14 0.14
0.14 0.14
0.14 0.14
B
0.28
0.24
0.20
0.24
0.26
0.26
0.25
0.23
0.20
0.17
0.15
0.14
0.13
0.13
C
0.30
0.23
0.23
0.25
0.26
0.27
0.25
0.22
0.20
0.17
0.15
0.14
0.14
0.13
D
0.69
0.31
0.19
0.24
0.26
0.26
0.25
0.23
0.20
0.17
0.15
0.14
0.14
0.14
33
DoT
n/a
0.12
0.12
0.16
n/a
0.19
0.16
0.11
0.08
0.07
n/a
0.04
n/a
0.04
Table 7. Replication of “Table 1 Means and Standard Deviations of Term Structure
Variables” (Campbell, 1995, page 135)
Panel A. Department of Treasury (DoT)
Variable
Excess return
Change in yield
Yield spread
2
0.074
(0.219)
-0.002
(0.254)
0.073
(0.119)
3
0.141
(0.407)
-0.001
(0.213)
0.140
(0.226)
Long bond maturities (months)
6
12
24
0.280
0.417
0.767*
(1.021)
(2.462)
(6.477)
-0.001
-0.001
-0.003*
(0.210)
(0.228)
(0.281)
0.274
0.401
0.697*
(0.341)
(0.391)
(0.569)
48
1.261*
(14.34)
-0.005*
(0.304)
1.023*
(0.785)
120
2.381*
(33.32)
-0.007*
(0.279)
1.528*
(1.182)
48
1.278*
(14.35)
-0.005*
(0.303)
1.023*
(0.809)
120
2.395*
(32.65)
-0.007*
(0.273)
1.512*
(1.239)
48
1.275*
(14.26)
-0.005*
(0.301)
1.018*
(0.807)
120
2.643
(32.42)
-0.008
(0.271)
1.732*
(1.273)
Panel B. Bloomberg (F082)
Variable
Excess return
Change in yield
Yield spread
2
0.000
(0.300)
-0.002
(0.247)
-0.001
(0.201)
3
0.098
(0.402)
-0.001
(0.203)
0.096
(0.240)
Long bond maturities (months)
6
12
24
0.277
0.443
0.722*
(1.002)
(2.566)
(6.454)
-0.001
-0.002
-0.003*
(0.205)
(0.236)
(0.280)
0.271
0.421
0.647*
(0.363)
(0.437)
(0.577)
Panel C. Federal Reserve Board (FRB)
Variable
Excess return
Change in yield
Yield spread
2
0.235
(0.307)
-0.001
(0.225)
0.236
(0.240)
3
0.251
(0.434)
-0.001
(0.206)
0.251
(0.264)
Long bond maturities (months)
6
12
24
0.313
0.460
0.751*
(0.957)
(2.488)
(6.360)
-0.001
-0.002
-0.003*
(0.199)
(0.229)
(0.276)
0.308
0.440
0.675*
(0.337)
(0.426)
(0.555)
Panel D. Weighted Svensson model (WSV) from the sample A (GovPx bond dataset)
Variable
Excess return
Change in yield
Yield spread
2
-0.022
(0.291)
-0.003
(0.249)
-0.025
(0.198)
3
0.041
(0.439)
-0.002
(0.223)
0.037
(0.224)
Long bond maturities (months)
6
12
24
0.196
0.416
0.731*
(0.955)
(2.479)
(6.368)
-0.001
-0.001
-0.003*
(0.201)
(0.228)
(0.276)
0.190
0.401
0.651*
(0.306)
(0.413)
(0.552)
48
1.260*
(14.42)
-0.005*
(0.305)
0.999*
(0.807)
120
2.606
(31.86)
-0.007
(0.267)
1.718
(1.273)
Panel E. Weighted Svensson model (WSV) from the sample B (GovPx bond dataset)
Variable
Excess return
Change in yield
Yield spread
2
-0.025
(0.291)
-0.003
(0.250)
-0.027
(0.196)
3
0.040
(0.439)
-0.002
(0.224)
0.036
(0.224)
Long bond maturities (months)
6
12
24
0.199
0.420
0.734*
(0.956)
(2.484)
(6.373)
-0.001
-0.001
-0.003*
(0.202)
(0.229)
(0.276)
0.193
0.405
0.654*
(0.308)
(0.416)
(0.551)
48
1.264*
(14.41)
-0.005*
(0.305)
1.002*
(0.808)
120
2.618
(31.85)
-0.007
(0.267)
1.726
(1.272)
48
0.475
(19.32)
0.014
(0.408)
1.141
(1.013)
120
-0.234
(36.77)
0.013
(0.307)
1.358
(1.234)
Panel F. Original Campbell’s Table 1
Variable
Excess return
Change in yield
Yield spread
2
0.379
(0.640)
0.014
(0.591)
0.196
(0.210)
3
0.553
(1.219)
0.014
(0.575)
0.324
(0.301)
Long bond maturities (months)
6
12
24
0.829
0.862
0.621
(2.950)
(6.203)
(11.29)
0.014
0.014
0.014
(0.569)
(0.546)
(0.486)
0.569
0.761
0.948
(0.437)
(0.594)
(0.799)
Campbell footnote: “Source: Author's calculations using estimated monthly zero-coupon yields, 1952-1991, from McCulloch
and Kwon (1993). The data are measured monthly, but expressed in annualized percentage points. Each row shows the mean
of the variable, with the standard deviation below in parentheses. Excess returns and yield spreads are measured relative to 1month Treasury bill rates.”
34
Table 8. Replication of “Table 2 Regression Coefficients” (Campbell, 1995, p.139)
Panel A. Department of Treasury (DoT)
Variable
Short-run changes
in long yields
Long-run changes
in short yields
2
1.207*
(0.296)
2.919*
(0.751)
3
0.492*
(0.144)
1.372*
(0.318)
Long bond maturities (months)
6
12
24
0.320*
0.168*
-0.003*
(0.090)
(0.073)
(0.056)
1.222*
1.528*
0.970*
(0.181)
(0.216)
(0.259)
48
-0.042
(0.040)
0.943*
(0.209)
120
-0.024
(0.022)
-0.074
(0.126)
48
-0.044
(0.038)
0.995*
(0.201)
120
-0.019
(0.021)
-0.069
(0.156)
48
-0.042
(0.038)
0.978*
(0.202)
120
-0.024
(0.020)
-0.069
(0.142)
Panel B. Bloomberg (F082)
Variable
Short-run changes
in long yields
Long-run changes
in short yields
2
0.202
(0.273)
0.949
(0.671)
3
0.378*
(0.133)
1.222*
(0.311)
Long bond maturities (months)
6
12
24
0.217*
0.123
-0.004
(0.074)
(0.067)
(0.055)
1.030*
1.185*
1.009*
(0.162)
(0.202)
(0.256)
Panel C. Federal Reserve Board (FRB)
Variable
Short-run changes
in long yields
Long-run changes
in short yields
2
0.301
(0.177)
1.397*
(0.457)
3
0.326*
(0.124)
1.193*
(0.276)
Long bond maturities (months)
6
12
24
0.260*
0.135*
-0.006
(0.078)
(0.067)
(0.057)
1.128*
1.243*
1.086*
(0.179)
(0.209)
(0.271)
Panel D. Weighted Svensson model (WSV) from the sample A (GovPx bond dataset)
Variable
Short-run changes
in long yields
Long-run changes
in short yields
2
0.872*
(0.352)
2.398*
(0.742)
3
0.667*
(0.198)
1.874*
(0.372)
Long bond maturities (months)
6
12
24
0.355*
0.154*
0.010
(0.095)
(0.072)
(0.058)
1.438*
1.376*
1.131*
(0.204)
(0.216)
(0.273)
48
-0.043
(0.039)
0.952*
(0.202)
120
-0.024
(0.020)
-0.074
(0.143)
Panel E. Weighted Svensson model (WSV) from the sample B (GovPx bond dataset)
Variable
Short-run changes
in long yields
Long-run changes
in short yields
2
0.880*
(0.358)
2.447*
(0.753)
3
0.648*
(0.194)
1.855*
(0.367)
Long bond maturities (months)
6
12
24
0.332*
0.145*
0.010
(0.092)
(0.071)
(0.058)
1.383*
1.328*
1.128*
(0.201)
(0.215)
(0.274)
48
-0.044
(0.038)
0.952*
(0.202)
120
-0.024
(0.020)
-0.074
(0.142)
48
-2.222
(1.451)
0.435
(0.398)
120
-4.102
(2.083)
1.311
(0.120)
Panel F. Original Campbell’s Table 2
Variable
Short-run changes
in long yields
Long-run changes
in short yields
2
0.019
(0.194)
0.510
(0.097)
3
-0.135
(0.285)
0.473
(0.149)
Long bond maturities (months)
6
12
24
-0.842
-1.443
-1.432
(0.444)
(0.598)
(0.996)
0.301
0.253
0.341
(0.147)
(0.210)
(0.221)
Campbell footnote: “Source: Author's calculations using estimated monthly zero-coupon yields, 1952-1991, from McCulloch
and Kwon (1993). Each row shows a regression coefficient β, with the standard error below in parentheses. Each coefficient
should be one if the expectations hypothesis holds. The regression in the first row is
)
(
)⁄(
where m is long bond maturity in months. The regression in the second row is
∑
(
)(
)
The standard error in the second row is corrected for serial correlation in the error term of the regression.”
35