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Empirical evidence indicates that both nominal and real yield curves in important markets have segmentation between their short end and their longer-maturity segments. This segmentation might affect term structure estimation, introducing distortions in longer-maturity yields, especially in parametric models. In order to deal with such segmentation, we propose a new model that combines the flexibility of spline functions with the parsimoniousness of a parametric four-factor exponential model. The short end of the yield curve is fitted using a B-spline function, while longer segments are captured by the parametric model. We illustrate the benefits of the proposed model for pricing and risk management purposes, using two examples: the real yield curve in the Brazilian government index-linked bond market, and the US Treasury nominal yield curve. We show that, in both markets, our model is simultaneously able to fit the yield curve well and to provide unbiased Value at Risk estimates for all tested portfolios of bonds, outperforming an important parametric benchmark model frequently adopted by central banks.
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A Hybrid Spline-Based Parametric Model for the
Yield Curve
Adriano Faria Caio Almeida
November 7, 2017
Abstract
Empirical evidence indicates that both nominal and real yield curves in impor-
tant markets have segmentation between their short end and their longer-maturity
segments. This segmentation might affect term structure estimation, introducing
distortions in longer-maturity yields, especially in parametric models. In order to
deal with such segmentation, we propose a new model that combines the flexibility
of spline functions with the parsimoniousness of a parametric four-factor exponen-
tial model. The short end of the yield curve is fitted using a B-spline function,
while longer segments are captured by the parametric model. We illustrate the
benefits of the proposed model for pricing and risk management purposes, using
two examples: the real yield curve in the Brazilian government index-linked bond
market, and the US Treasury nominal yield curve. We show that, in both markets,
our model is simultaneously able to fit the yield curve well and to provide unbiased
Value at Risk estimates for all tested portfolios of bonds, outperforming an impor-
tant parametric benchmark model frequently adopted by central banks.
Keywords: Spline Models, Exponential Term Structure Models, Curve Fitting,
Risk Management, Price Index.
JEL Code: C51, C58, G17.
The second author acknowledges financial support from CNPq.
Email: afaria@fgvmail.br, FGV EPGE Brazilian School of Economics and Finance, Rio de Janeiro,
Brazil.
Email: calmeida@fgv.br, FGV EPGE Brazilian School of Economics and Finance, Rio de Janeiro,
Brazil.
1 Introduction
Understanding the term structure of interest rates is fundamental for investors, regulators,
and risk managers. It contains valuable information about monetary policy, interest rate
risk factors, and fixed-income trading decisions. To identify, and possibly anticipate,
the dynamics of interest rates across different fixed-income markets, researchers have
proposed several distinct methods to estimate the term structure. The yield curves
generated by these methods serve different purposes and end up reflecting idiosyncrasies
of each particular market analyzed.
A technical report produced by the Bank for International Settlements (BIS (2005))
presents the main term structure models adopted by major central banks, classifying
them as parametric and spline-based models. Parametric models fit the yield curve in
a parsimonious way. They smooth local idiosyncrasies, such as hedging demand effects,
liquidity, and inflation risk premia, in favor of a common global behavior. These methods
are typically used in macroeconomic studies, in which smoothness and the ability to
capture common movements are at least as important as model accuracy. Some important
benchmarks in this class include the three-factor exponential model of Nelson and Siegel
(1987), its four-factor extension proposed by Svensson (1994), and their corresponding
dynamic extensions proposed by Diebold and Li (2006) and de Pooter (2007).
Unlike parametric models, spline-based models do not impose parsimonious functional
forms along the entire length of maturity. Instead, they are constituted by several low-
order polynomials which are smoothly linked over the range of maturities. Therefore,
when compared to parametric models, splines have a larger number of parameters and
less smooth yield curves. On the other hand, they are more accurate and have higher
ability to fit idiosyncrasies. As noted by Gurkaynak et al. (2007), a trader looking for
pricing anomalies can be quite concerned about how a particular bond is priced, relative
to others nearby. In this sense, less smoothness and greater accuracy in curve regions
with many idiosyncrasies are clearly desirable. Within the spline-based class, we can
highlight the initial works of McCulloch (1971), McCulloch (1975) and Vasicek and Fong
(1982) and more recent extensions including the penalized spline models of Fisher et al.
(1995), Waggoner (1997) and Jarrow et al. (2004).
An important drawback of the parametric approach is the instability of the solution
in response to idiosyncratic shocks. As illustrated by Anderson and Sleath (2001), a
1
small perturbation in the price of just one bond can significantly affect an estimate of
the entire curve. The spline-based models avoid this undesirable characteristic. Since
each polynomial in a spline behaves like a local function, spline-based models can easily
accommodate local shocks in any specific curve region.1
There is compelling empirical evidence documenting the existence of segmentation
between the short end of the yield curve and its longer-maturity segments. This is true
for both nominal and real markets, and across different countries. Such segmentation
may decrease the accuracy (in estimation) of benchmark parametric term structure mod-
els adopted by central banks, such as those of Nelson and Siegel (1987) and Svensson
(1994). Therefore, there is a need for better alternative models for central banks and
other researchers interested in the simplicity and parsimoniousness of parametric models,
without sacrificing pricing and hedging accuracy. In this paper, we aim to fill this gap.
For term structure segmentation, Ejsing et al. (2007) document that the short end
of the real term structure is subject to the erratic and seasonal behavior of the price
index. Such behavior does not affect its medium-term and long-end segments, since the
seasonality effect decreases over maturities. When considering the nominal yield curve,
Knez et al. (1994) and Greg (1996) identify a common component in the US Treasury bill
yields that is not related to the yields of longer-maturity notes and bonds. Piazzesi (2005)
suggests that parametric latent factor models of the yield curve, not taking into account
monetary policy actions, have poor fitting in particular for short-term maturities. This
is true because monetary policy actions such as FOMC meetings make the short end of
the curve behave in a particular way, not always shared by longer-maturity yields.2
Those short-end patterns, added to the high degree of smoothness and instability of
the parametric models, imply at least two main problems. First, these models are not able
to adjust to the idiosyncrasies of the short end of the curve, especially for the real case, as
shown by Ejsing et al. (2007). Second, when attempting to fit this short end, the model
can distort estimates of the long end of the curve. Regarding this last issue, Anderson
and Sleath (2001) show that money market rates can be used in a nonparametric model
1As an example of possible advantages of this local behavior, Almeida et al. (2017) show that seg-
mented exponential-based spline models present superior ability, relative to parametric models, in fore-
casting the short end of the US yield curve due to their success in capturing idiosyncratic shocks.
2A more general form of segmentation manifests itself through demand and supply factors affecting
specific regions of maturities of the nominal terms structure of interest rates. For examples, see Green-
wood and Vayanos (2010), Greenwood and Vayanos (2014), and Krishnamurthy and Vissing-Jorgensen
(2012).
2
to significantly improve estimates of the short end of the UK nominal term structure.
However, using money market rates to improve Svensson’s estimates affects the fitted
yields at the very long end.
One possible general procedure, which can be applied to both nominal and real yield
curves, is to determine a threshold such that securities with maturity below it are elimi-
nated in the estimation process.3For example, Gurkaynak et al. (2010) set this threshold
at eighteen months, to estimate the US real yield curve using Svensson’s model. How-
ever, as stated by the Deutsche Bundesbank in BIS (2005), extracting the term structure
without enough data at the short end may generate unrealistic estimates not only for
this part of the curve, but also for the region right after the threshold. Additionally,
this lack of data may cause instability in the time series of a model’s parameters, usually
generating unrealistic volatility patterns. Those issues compromise a model’s usefulness
in dynamic applications, such as forecasting exercises and risk management procedures.
All in all, despite their instability issues, parametric models have advantages over
spline-based ones when parsimony is required, especially in dynamic applications. Their
reduced number of parameters, between four and six, is crucial to estimate dynamic
volatility models. These models are used to perform risk management tasks in interest
rate markets, for example, to compute Value at Risk (VaR) and elaborate hedging strate-
gies. Parsimony is also an essential characteristic in the dynamic version of the Nelson
and Siegel model proposed by Diebold and Li (2006). It has been applied to forecasting
the yield curve and to modeling dynamic interactions between the macroeconomy and
the yield curve (see, for instance, Bianchi et al. (2009) and Diebold et al. (2006)).
This paper provides a unique approach capable of dealing with the short-term issues
and their impact on estimation of the whole yield curve, while simultaneously keeping
the desirable parsimony necessary in dynamic applications. We propose a segmented
model for the yield curve, considering splines to fit the short end, while the medium-
term segment and long end are captured by a parametric Svensson model. The proposed
model is useful for pricing purposes and monetary policy analysis. At the same time,
the parsimonious parametric form for longer maturities of the yield curve enables the
3Trying to circumvent instability issues on the real yield curve, Ejsing et al. (2007) suggest an approach
in which the seasonality effect from bond prices is extracted before proceeding to curve estimation using
the Nelson and Siegel model. However, this methodology is not suitable to price index-linked bonds,
since pricing these instruments requires taking into account inflation accrual, which is directly affected
by seasonality.
3
construction of economically interpretable risk controls via VaR and stress scenarios.
The main challenge of the proposed approach consists in solving a nonlinear opti-
mization problem where two distinct models are jointly estimated. The nonlinearity is
inherited from the non-linear relationship between bond prices and the term structure of
interest rates. In the estimation process, restrictions must be imposed to guarantee the
smoothness of the segmented yield curve. We develop a procedure in which the restricted
problem is rewritten as an unconstrained problem that satisfies all the restrictions of
the first one. The first step consists in specifying the full model as a spline function.
Following that, for the medium and long terms, we project the functional forms of the
parametric model onto the spline basis, obtaining a new basis of functions, which we
denominate “mixed basis.” This basis has the nice property of behaving like a spline
model in the short end and like a parametric model for longer maturities. Therefore, our
method provides a single basis with the desired segmentation (the mixed basis), allowing
for a direct and unrestricted estimation of the parameters associated with it.
We evaluate the performance of our proposed approach in both nominal and real
cases. We apply our model to the US nominal government security market for the pe-
riod from January 1995 to December 2016, and to the Brazilian government index-linked
bond market, covering January 2009 to May 2014. In the US nominal yield curve, we
are particularly interested in investigating if Treasury bills with their documented local
component can affect the whole term structure estimation process. To analyze the sea-
sonal effect in the real yield curve estimation, we select the Brazilian market, in which
differences between short-term inflation rates for different terms are above average when
compared to other important markets.4For example, in the period of 2010 to 2016, the
mean absolute difference between the annualized inflation rates for six months and one
year was of 160 basis points (bps).5Considering that the difference between nominal
yields for six months and one year is usually a few bps, the discrepancy from the respec-
tive real yields will be around 160 bps as well. Consequently, a model must have enough
flexibility on the short end to accommodate such huge variation in a small-term interval.
4According to data extracted from Barclays Universal Government Inflation-Linked Bond Index
(UGILB), and Barclays Emerging Markets Government Inflation Linked Bond Index (EMGILB), in
January 2014 the Brazilian market was one the five largest in the world and the largest among all emerg-
ing countries, totaling U 220 billion. For comparison purposes, the largest inflation market is the TIPS
one, and it had U 959 billion outstanding at the same date.
5This difference for the US, UK and French markets was of 137, 74 and 92 bps, respectively.
4
We find that the Svensson model provides an average good fit to the US nominal
market. However, short-term securities end up impacting yield estimates at the long end,
and compromise estimates for the daily realized volatility of yields variations (first dif-
ference in yields). In clear contrast, our hybrid model does not present those undesirable
results. In addition, as a consequence of the worse fitting to daily volatility provided by
the Svensson model when compared to our hybrid model, our proposed design provides
better ranked VaR forecasts for different US Treasury portfolios.
Finally, regarding the Brazilian real market, parametric models cannot provide reli-
able estimation of the real term structure when the short end is populated with short-term
instruments. Moreover, their VaR estimates are biased when we extract the term struc-
ture without short-term securities. On the other hand, our hybrid model is able to not
only fit the real yield curve as a whole, but also to provide unbiased estimates of VaR for
various portfolios holding inflation-linked bonds.
The paper proceeds as follows. Section 2 presents the two classes of models, as well as
the proposed segmented design. This section also contains a numerical example, which
illustrates how we build the segmented basis. Section 3 introduces the data sets for the
US and Brazilian markets. Section 4 presents results of the two empirical applications,
considering the two different data sets. For each data set, the first application compares
in-sample and out-of-sample fitting abilities across different static term structure models.
The second consists of VaR estimation for portfolios exposed to nominal and real term
structure risk. Section 5 concludes.
2 The Model
We start by analyzing the case in which the term structure of interest rates can be approx-
imated by a general functional basis. After providing this general structure, we formally
define the Svensson and the B-spline bases representing, respectively, the parametric and
the spline-based approaches. Merging these two approaches, we describe the methodology
used to build the mixed basis, which defines our segmented model. We end this section
by providing a numerical example in which we explain how to construct the proposed
mixed basis.
5
2.1 The general case
Consider a domain [0, T ] in which the term structure of interest rates z(.) is defined. Let
Nbe the number of bonds, pibe the price of bond i,ci= (ci1, ci2, ..., cimi) be the vector
of cash flows with mipayments for bond i, and τi= (τi1, τi2, ..., τimi) be the vector of
maturities of those payments. In this context, the exact relationship between piand z(.)
is given by
pi=
mi
X
j=1
cijexp(τijz(τij)), i = 1,2, ..., N. (1)
In order to propose a model to approximate the term structure, we consider a certain
basis δ(.) = (δ1(.), δ2(.), ..., δκ(.)), with dimension κ, which approximates z(.) via a linear
combination of elements in δ(.). Using equation (1), we can express the price of a bond
as the following function of δ:
pi=
mi
X
j=1
cijexp(τijδ(τij)β) + i=ciexp(Diδ(τi)β) + i,
where βis a κ×1 parameter vector, iis an error term generated by the approximation
of z(.) by δ(.), Diis a diagonal matrix whose main diagonal is given by the vector of
maturities τi, and
δ(τi) = (δ(τi1)|, δ(τi2)|, ..., δ(τimi)|)|,
is a mi×κmatrix that results from applying δto each element of τi. Hence, the theoretical
price of bond iproduced by this model is given by
πi=ciexp(Diδ(τi)β).(2)
Estimates of the parameters βmay be obtained by solving
min
β(α) N
X
i=1 piπi(β(α))
wi2
+ZT
0
α(τ)h00(τ)2 !,(3)
where wiis the duration of bond i,h(τ) = δ(τ)βand α(.) is a penalty function such that
α: [0, T ]R+.
The first part of the objective function in (3) represents the sum of quadratic pricing
6
errors, weighted by the inverse of each bond’s duration.6It represents the component
related to model fitting. The second part of the objective function penalizes excess
variability in the estimated term structure. This is the component associated with model
smoothness.
For the parametric and standard spline models, α(.) is simply null. In this case, the
degree of smoothing in the approximation of the term structure is determined, a priori,
by the chosen basis δ(.). On the other hand, for penalized spline models, α(.) can take
any positive value. For these models, the solution of problem (3) must consider a trade-
off between fitting and smoothness, with the importance of the latter determined by the
magnitude of α(.). In the extreme case when α→ ∞, any minimal degree of curvature
gets an infinite penalty. Then, the solution will converge to a specific linear combination
of the elements in δ(.) that approximates a line. As an example, for the cubic spline
basis, when α→ ∞, the solution converges to the least-squares line.
The functional form adopted for α(.) differs across various penalized spline models.
Fisher et al. (1995) let α(.) be constant over the entire domain of maturities, and deter-
mine its value by the Generalized Cross Validation method (GCV). For Waggoner (1997),
α(.) is a function that assumes different values over the range of maturities. Due to such
variability of the penalty function, this method is denominated VRP, for Variable Rough-
ness Penalty. The main argument for having a variable penalty function across maturities
is the stylized fact that fluctuations in the long end of the curve are much less volatile
than those in the short end. Therefore, imposing the same degree of smoothing over the
entire curve will not be appropriate. The parameters appearing in α(.) are estimated
using an out-of-sample procedure.
The component associated with model smoothness can be written as
ZT
0
α(τ)h00(τ)2=β|ZT
0
α(τ)δ00(τ)|δ00(τ)β=β|H(α)β,
6The price piof bond iis given by the folowing function of its yield to maturity or internal rate of
return yi:pi=Pmi
j=1 cijeyiτij. The duration wiis given by minus the first derivative of price with
respect to yield to maturity: wi=dpi
dyi. Using these two concepts, for both observed and modeled
prices, we can calculate a first order Taylor approximation of the error in price with respect to the error
in yield to maturity obtaining: piπi≈ −wi(yiyπ
i), where yπ
irepresents model-implied yield to
maturity. Using this approximation, Vasicek and Fong (1982) show that, in term structure of interest
rates models, which assume that the cross-section of yields is homoskedastic, weighting the pricing error
of each bond by the inverse of its duration generates a set of cross-sectional bond pricing errors that are
approximately homoskedastic.
7
where His a matrix κ×κ, in which element Hi,j is defined as RT
0α(τ)δ00
i(τ)δ00
j(τ). Using
the last equation, we can rewrite the minimization problem (3), for a given α, as follows:
min
β(α)PΠ(β(α))|WPΠ(β(α))+β(α)|H(α)β(α),(4)
where Pis an N×1 vector of bond prices {pi}i=1,...,N , Π(β) is the corresponding vector
of modeled prices, {πi(β)}i=1,...,N , and Wis an N×Ndiagonal matrix with Wi,i =
(1/wi)2, i = 1, ..., N .
The minimization in (4) can be solved as a nonlinear least squares problem. Fisher
et al. (1995) linearize Π(β) around an initial guess β0:
Π(β)Π(β0) + Π(β)
∂β|(ββ0),(5)
where Πi(β)
∂β|=πi(β)ciDiδ(τi). Defining X(β0) = Π(β)
∂β|β=β0and Y(β0) = PΠ(β0) +
X(β0)β0, and rearranging (4) yields
min
β(α)Y(β0)X(β0)β(α)|WY(β0)X(β0)β(α)+β(α)|H(α)β(α).(6)
For a given α, the minimizer for (6) is
β1(α) = X(β0)|W X (β0) + H(α)1X(β0)|W Y (β0),
where β1(α) is an updated β0. One can use β1(α) as an initial guess for the next iteration,
obtaining β2(α), and continue iterating until convergence. The solution is the following
fixed point:
β(α) = X(β(α))|W X (β(α)) + H(α)1X(β(α))|W Y (β(α)).(7)
2.2 Svensson’s basis
The Svensson basis is the vector ψ={ψk}4
k=1, which is constituted by four functions:
level, slope and two curvatures. For any given, strictly positive and distinct, λ1and λ2,
the basis is defined as
8
ψk(τ) =
1 , if k= 1
S(λ1, τ ) = (1 exp(λ1τ))/(λ1τ) , if k= 2
C(λ1, τ ) = S(τ, λ1)exp(λ1τ) , if k= 3
C(λ2, τ ) = S(τ, λ2)exp(λ2τ) , if k= 4
.(8)
The Nelson and Siegel basis is obtained by eliminating the second curvature ψ4(τ)
of the Svensson basis. To specify and implement the parametric model, we have to
substitute the generic basis δ(.), appearing in the general case, by the specific parametric
basis ψ(.). Both the Nelson and Siegel and the Svensson models have been extensively
adopted for pricing, hedging and monetary policy purposes (see Diebold and Rudebusch
(2013)).
2.3 B-spline basis
Let {sk}K
k=1 denote the set of knot points, with sk< sk+1,s1= 0, and sK=T. The
spline of degree pis defined as
S(τ) =
p
X
j=0
θjτj+
K1
X
k=1
ηk|τsk|3,
for some constants {θ}p
j=1 and {η}K1
j=1 and for τ[0, T ]. Following most studies in the
literature, we adopt a cubic spline, therefore fixing p= 3.
A stable numerical parameterization of a spline is provided by a B-spline basis. Let
{dk}K+6
k=1 denote the augmented set of knot points, with d1=d2=d3=s1,dK+4 =
dK+5 =dK+6 =sKand dk+3 =skfor 1 kK. A B-spline of degree pis defined by
the following recursion:
φp
k(τ) = φp1
k(τdk)
(dk+pdk)+φp1
k+1(τ)(dk+p+1 τ)
dk+p+1 dk+1
,(9)
for k∈ {1, ..., K + 5 p}and τ[0, T ], with
φ0
k(τ) =
1 if dkτ < dk+1
0 c.c.
.
A B-spline basis of degree 3 is a vector of κ=K+ 2 cubic B-splines given by
9
φ3(τ) := (φ3
1(τ), ..., φ3
κ(τ))1×κ. To simplify notation, let φk(τ) := φ3
k(τ) and φ(τ) := φ3(τ).
As shown by De Boor (2001), any cubic spline can be represented as a linear combination
of the B-splines that constitute the basis φ(.). That is, S(τ) = Pκ
k=1 βkφk(τ), with βkR
k[1, κ].
We follow the general case presented in Section 2.1 to specify and implement a cubic
B-spline model, substituting for the generic basis δ(.), the specific B-spline basis φ(.).
2.4 Mixed basis
Among the existing properties of the B-spline basis (see De Boor (2001) and Lyche and
Morken (2011)) three are particularly useful to construct our proposed mixed basis. (i)
Partition of unity: for any τ[0, T ], Pκ
k=1 φk(τ) = 1. (ii) Local knots: the kth B-spline
φkdepends only on the knots dk,dk+1, ..., dk+4. (iii) Local support: the interval [dk, dk+4]
is denominated support of φk. For any k(1, κ), if τ(dk, dk+4), then φk(τ)>0;
otherwise, φk(τ) = 0.
Using the last two properties, if we split the domain [0, T ] into two intervals, [0,¯τ]
and (¯τ , T ], only a subset of the κB-splines composing the basis will take non-zero values
over the last interval. Formally, let ¯τ(0, T ) and d¯
kbe the greatest knot in (0,¯τ],
which implies that ¯
k∈ {5,6, ..., κ 1}. Therefore, the set formed by B-splines that takes
non-zero values in the range (¯τ, T ] is given by
¯
φ:= (φ¯
k3, φ¯
k2, ..., φκ),(10)
containing ¯κ=κ¯
k+ 4 elements. We denominate ¯
φas the active B-spline basis, with
dimension ¯κ, over (¯τ, T ].
To construct the mixed basis, we start with a B-spline basis φ, defined over the
domain [0, T ], containing κelements. First, we determine the range, (¯τ, T ], over which
the term structure will be modeled using the Svensson basis (parametric basis). We then
approximate, in (¯τ , T ], the smooth functions that constitute this parametric basis, by
using the active B-spline basis ¯
φ. Note that this approximation uses only a subset of the
κB-splines (the active ones), leaving the others free to capture the short end of the term
structure. A natural question arises. How many knot points, or equivalently how many
B-splines ¯
φ, active within the interval (¯τ , T ], should exist to make this approximation of
10
the parametric model work? Lyche and Morken (2011) show that it is possible to derive
local error bounds for the spline approximation. Those bounds are functions of the
spacing between the knot points and the degree of the spline, as happens in polynomial
approximations. In the appendix, we present such bounds and an algorithm which uses
them to determine an optimal set of knots’ locations within (¯τ, T ]. For now, assume that
¯
φhas enough knot points to keep the approximation error in the Svensson basis within
an acceptable tolerance.
Considering that the parametric Svensson model should be active only for maturities
τ > ¯τ, we use a translation to redefine its basis to be
¯
ψk(τ) =
1 , if k= 1
S(λ1, τ ) = (1 exp(λ1(τ¯τ)))/(λ1(τ¯τ)) , if k= 2
C(λ1, τ ) = S(τ, λ1)exp(λ1(τ¯τ)) , if k= 3
C(λ2, τ ) = S(τ, λ2)exp(λ2(τ¯τ)) , if k= 4
.(11)
Therefore, the approximation of the transformed parametric basis ¯
ψk(.) by the active
B-spline basis ¯
φ(.) over the interval (¯τ , T ] is given by
¯
ψk(τ) = ¯
φ(τk+k(τ), k ∈ {1,2,3,4}
where kis the error term and Γkis a ¯κ×1 vector of parameters.
By the partition of unity property, making Γ1a vector with all entries equal to one,
we get ¯
ψ1(τ) = ¯
φ(τ1. Thus, ¯
ψ1(.) is perfectly replicated by ¯
φ. In turn, for all other
functions that constitute the transformed parametric basis ¯
ψ, the parameter vector Γk
can be obtained by regression splines:
Γk= ( ¯
φ|¯
φ)1¯
φ|¯
ψk,
where ¯
φand ¯
ψkresult from applying ¯
φ(.) and ¯
ψk(.) to each observed maturity within
the interval (¯τ, T ]. For those cases, kis not null, and depends on the spacing between
the knot points.
Defining the ¯κ×4 matrix Γ := (Γ1,Γ2,Γ3,Γ4), the approximation of the Svensson
basis ¯
ψ(.), defined over the interval (¯τ , T ], is provided by
11
ˆ
ψ(τ) = ¯
φ(τ.(12)
Given (10) and (12), the mixed basis Ψ(.), defined over the whole domain [0, T ], is given
by
Ψ(τ) = (φ1(τ), ..., φ¯
k4(τ))1×(¯
k4) ˆ
ψ(τ)
=(φ1(τ), ..., φ¯
k4(τ))1×(¯
k4) ¯
φ(τ)
I(¯
k4) 0(¯
k4)×4
0¯κ×(¯
k4) Γ
κׯ
k
=φ(τ)A, (13)
where Ijis the identity matrix with dimension j, 0i×jis a i×jnull matrix, and
A=
I(¯
k4) 0(¯
k4)×4
0¯κ×(¯
k4) Γ
κׯ
k
.
The Ψ basis has dimension ¯
k, with four functions used to approximate the Svensson basis,
and ¯
k4 representing free B-splines.
By (13), the mixed basis, Ψ(.), is a linear function of the B-spline basis φ(.), with
linear weights determined by matrix A. In fact, matrix Adetermines the dimension of
Ψ(.), the number of free B-splines, and the lowest maturity from which the mixed basis
approximates the Svensson basis. Therefore, matrix Arestricts the B-spline basis in order
to build the desired mixed basis.
Seen from another perspective, it is also possible to find a matrix Msuch that the
product between Mand the vector of parameters associated with the B-spline basis makes
the models that use the B-spline basis and the mixed basis equivalent to each other. First,
using φ(.) and Ψ(.) in the pricing equation (2), we get
12
πSp
i(βSp) = ciexp Diφ(τi)βSp (14)
πM
i(βM) = ciexp DiΨ(τi)βM
=ciexp Diφ(τi)M(15)
=πSp
i(βSp =M),(16)
where, πM
i(πSp
i) is the modeled price for the bond iobtained using the mixed (B-spline)
basis and the associated parameter vector βM(βSp). By equation (16), the two bases will
be equivalent when βSp =M. Using that the columns of Aare linearly independent,
the relationship between βMand βSp can also be expressed by7
βM= (A|A)1A|βSp.(17)
Substituting (17) in (15),
πM
i(βM) = ciexp Diφ(τi)MβSp ,(18)
where
M=A(A|A)1A|
=
I(¯
k4) 0(¯
k4)ׯκ
0¯κ×(¯
k4) Γ(Γ|Γ)1Γ|
κ×κ
.
Finally, comparing the modeled price using splines (14) to the modeled price using
the mixed basis (18), we observe that the only difference is the presence of matrix Min
(18). This matrix linearly combines parameters in βSp to achieve the equivalence between
using the B-spline basis or the mixed one. Nevertheless, since the first ¯
k4 elements of
the mixed basis are free B-splines, Mdoes not restrict the first ¯
k4 parameters of βSp.
Matrix Ain (13) and matrix Min (18) represent linear transformations applied,
respectively, to a subset of elements of φ(.), and to a subset of associated parameters
7The linear independence of Γ’s columns directly implies the linear independence of A’s columns.
Γ’s columns are linearly independent because λ16=λ2guarantees that the exponential functions that
compose the Svensson basis are all linearly independent.
13
βSp. Such transformations guarantee that the resulting mixed model approximates the
Svensson basis for maturities above a particular ¯τin the domain. Since the Svensson
basis provides a higher degree of smoothness than the B-spline basis, the above-mentioned
transformations produce different levels of smoothness over the domain. Therefore, the
model resulting from the use of the mixed basis Ψ(.), is called VRP-SV. The term “VRP”
comes from the non-uniformity of smoothness across maturities, and the term “SV” comes
from the approximation of the Svensson model for maturities above ¯τ.
2.5 Numerical example
We provide a graphical representation of how the mixed basis is built. Consider the do-
main [0,10] and a cubic spline characterized by the set of knot points s={0,1, ..., 9,10},
containing 11 elements. The B-spline basis capable of representing this spline is obtained
by (9), using the augmented set of knot points {dk}11+6
k=1 , where d1=d2=d3=s1,
d11+4 =d11+5 =d11+6 =s11 and dk+3 =skfor 1 k11. This basis contains κ= 13
B-splines, represented in Figure 1a.
012345678910
-0.5
0
0.5
1
τ
Base B-Spline φ(.)
012345678910
-0.5
0
0.5
1
τ
Base Mista Ψ(.)
(a) B-spline basis φ(.)
012345678910
-0.5
0
0.5
1
τ
Base B-Spline φ(.)
012345678910
-0.5
0
0.5
1
τ
Base Mista Ψ(.)
(b) Mixed basis Ψ(.)
Figure 1: Bases
14
Choosing arbitrarily ¯τ= 2 and noting that ¯
k= 6 and d¯
k= 2, we define the
mixed basis. By (10), the active B-spline basis over the interval (¯τ, 10] is given by
¯
φ= (φ3, φ4, ..., φ13). In Figure 1a, the active basis ¯
φis represented by the set of B-splines
plotted with a solid line. For given values of λ1and λ2, the Svensson basis ¯
ψ, as defined
in (11), is approximated by the basis ¯
φusing the projection appearing in (13). As a result
of this process, we obtain the mixed basis Ψ(.), plotted in Figure 1b. This basis is formed
by six functions, from which the first ¯
k4, i.e. two, functions are the B-splines that
assume null value for maturities above ¯τ. The other four functions approximate those
that constitute Svensson’s basis ¯
ψ, for maturities above ¯τ.
This approximation becomes more accurate with smaller spacing between the knot
points. In fact, the number of knot points above ¯τused in this example is not sufficient
to provide an accurate approximation. We used a small number of knot points to provide
a better graphical visualization. In the appendix, we show a histogram of the set of knot
points obtained based on our knot location algorithm that guarantees a small error in
the approximation.
Considering the B-spline and the mixed bases created above, we apply the cubic spline
and VRP-SV models to a given series of simulated data. We consider four levels for the
penalty parameter α, as shown in panels (a) and (b) of Figure 2. Note that when α= 0,
the proposed method can fit strong movements in the short end without compromising
the fitting in other parts of the curve. In fact, the estimates of the VRP-SV model
resemble those obtained by the flexible cubic spline method, except when α→ ∞. In
that case, the cubic spline solution converges to the least-squares line, while the estimate
of the proposed model goes to a constant. This happens because the only linear design
accepted by Svensson’s basis, and consequently by the mixed basis, is the one obtained
giving nonzero weight only to the level function (ψ1).
Finally, we check the performance of our method to approximate the Svensson model
above ¯τ. To do this, we calculate the approximation error for the VRP-SV model, defined
as
max
τ[0,T ]Ψ(τ)¯
Ψ(τ)β(α),(19)
where ¯
Ψ(τ) = (φ1(τ), ..., φ¯
k4(τ))1×(¯
k4) ¯
ψ(τ). For αequal to 0, 1, 1e+2 and 1e+8,
the errors are 2e-8, 1e-8, 2e-9 and 6e-13, respectively. The error is negatively correlated
15
with αbecause when αgrows, the solution converges to the level function, which is
perfectly replicated by the B-spline basis.
Figure 2: Fitting
3 Data
This section provides a description of the Brazilian and US data sets used for curve
estimation in the subsequent empirical section. First, we briefly describe the Brazilian
government index-linked bond market, explain an issue of scarcity of short-term data
existent in this market, and provide a possible solution to deal with this issue. Second,
we detail data period, frequency, and type of US Treasury instruments selected for the
two empirical applications: term structure fitting and risk management.
3.1 Brazilian government index-linked bonds data set
The main Brazilian Treasury inflation-protected security is the NTN-B bond. Coupon
interest on these bonds is paid twice a year, with an annualized rate of 6% adjusted for
changes in the leading consumer price index (IPCA). 8IPCA is the index used for inflation
8Since any NTN-B pays in the middle of a month, and since inflation accruals use the price index
of the previous month, there is an indexation lag of fifteen days when this security is priced by market
participants. Such a lag exists in all index-linked bond markets across the world, and the lag in the
Brazilian market is considerably shorter than in other countries. For instance, in the TIPS and Gilts
markets, there are indexation lags of 2.5 and 3 months, respectively. Since the lag is short in the NTN-B
market, we abstract from this indexation issue when estimating the term structure. For those interested
in a methodology to deal with this indexation issue see Evans (1998).
16
(a) Set of maturities (b) Set of maturities - zoom
Figure 3: Bond maturities
targeting regimes by the Central Bank of Brazil. Prices on the secondary market are
published daily by the Brazilian Financial and Capital Markets Association (ANBIMA).
They are obtained by averaging bond prices collected from main market participants who
send their estimated fair prices to ANBIMA, in a process similar to how the LIBOR rate
is calculated.
The set of maturities of NTN-B bonds available in our sample is depicted in Figure
3a. Each bond is represented by a solid line, which associates each day in the period
(horizontal axis) to its remaining years to maturity (vertical axis). At any point in time,
despite the small number of available bonds, there is a wide range of maturities, from 0
to 40 years. Note that this market structure implies a considerable maturity gap between
subsequent bonds. For yield curve estimation, such a gap in the short end is of particular
concern, since it may require model extrapolations. In particular, during some periods,
there is no available bond with time to maturity of less than one year. For example, this
was the case in 2013, when the shortest maturity available was of 1.25 years, as shown in
Figure 3b.
The lack of available data at the short end of the yield curve is more severe whenever
there are no trading prices for bonds close to maturity, or when those prices are affected by
idiosyncratic premia, such as the liquidity premium. Unfortunately, this is the case for the
17
jan/10 jan/11 jan/12 jan/13 jan/14
0
2
4
6
8
10
annual rate %
monthly semiannual annual
jan/10 jan/11 jan/12 jan/13 jan/14
0
2
4
6
8
10
annual rate %
monthly semiannual annual
(a) Inflation
jan/10 jan/11 jan/12 jan/13 jan/14
0
2
4
6
8
10
annual rate %
monthly semiannual annual
jan/10 jan/11 jan/12 jan/13 jan/14
0
2
4
6
8
10
annual rate %
monthly semiannual annual
(b) Forecasted inflation
Figure 4: Actual and Forecasted Inflation
NTN-B government bond market. Its bottom range of maturities is constituted of bonds
issued with medium or long duration that over time become bonds close to maturity. As
reported by most central banks in BIS (2005), such bonds are not useful to obtain proper
references for the short end of the curve, since they are usually highly illiquid. In cases
like this, the solution adopted by most central banks consists of eliminating bonds with
time to maturity less than one year from the estimation procedure, and instead, using
bills and money market rates, when available, to populate the short end.
In the Brazilian market, there is a small number of established references available for
the short end of the real yield curve. Therefore, when we estimate the real term structure,
we try to exclude the smallest possible number of short-term bonds. Nonetheless, we
cannot avoid excluding some highly illiquid short-term bonds. In fact, as shown in Figure
4a, the strong seasonal pattern of the price index (IPCA) and its large variability for
terms below six months represent important issues when pricing real bonds, since these
variations are usually barely reflected in prices of available illiquid short-term bonds. In
line with that, many times it is possible to observe unrealistic behavior for bonds below
six months, such as smooth price patterns or even almost flat yields over months. For
this reason, we exclude bonds with time to maturity of less than six months, and seek
alternative ways of anchoring the short end of the real yield curve.
Estimating the term structure without enough data at the short end may cause sev-
18
eral problems. As stated by the Deutsche Bundesbank in BIS (2005), it generates un-
realistic short end estimates, and sometimes even unrealistic estimates for the one-year
zero-coupon yield, which has particular importance for policymakers. Moreover, it causes
instability in the time series of models’ parameters, usually generating unrealistic volatil-
ity patterns. These issues compromise models’ usefulness in dynamic applications, such
as forecasting exercises and risk management procedures.
Seeking to provide more realistic estimates and to preserve models’ usefulness in
dynamic applications, we construct synthetic short-term bonds to populate the short
end. These bonds are built by discounting nominal zero-coupon rates by market inflation
expectations for one-, three- and six-month maturities. On every business day, ANBIMA
publishes estimates for the nominal yield curve extracted from the Brazilian Treasury
nominal bonds.9For inflation expectations, we use data from the Focus Market Readout,
a weekly survey of market forecasts for the main Brazilian macroeconomic variables,
released by the Central Bank on Mondays.10 Figure 4bshows a time series of forecasted
inflation over our sample period. Compared to Figure 4a, it is possible to see that the
forecasted inflation resembles the actual inflation, with the same seasonal patterns.
Our sample covers the period from January 2009 to May 2014, totaling 1332 days of
data on NTN-B and synthetic bonds.
3.2 US Treasury data set
The US nominal yield curve is extracted using Treasury bills, notes and bonds. Following
Bliss (1996) and Waggoner (1997), we take the price quotes data set from CRSP Govern-
ment Bond files, excluding bills with less than 30 days to maturity, notes and bonds with
less than a year to maturity, flower and callable bonds. For our first empirical applica-
tion, yield curve fitting, we use month-end prices from January 1995 to December 2016,
covering a total of 264 months. For our second application, risk management analysis,
we use daily quoted prices from January 2010 to December 2016, totaling 1752 days.
For the period from January 1995 to December 2016, on the last day of each month
9There is no issue of scarcity of short-term data in the nominal yield curve simply because nominal
bonds are more liquid than real bonds. In fact, there are nominal bonds issued with short time to ma-
turity, the maturity gap between subsequent bonds is small, and there are available short-end references
coming from money market rates.
10In the Brazilian market, the use of inflation expectations based on the Focus Market Readout is a
consensus among newspapers, analysts, and policy makers. In addition, it is shown by Ang et al. (2007)
that surveys have higher predictive value than traditional models in forecasting inflation expectations.
19
there are, on average in the cross-section, 170 securities and not less than 100. These
Treasury securities cover a maturity range from one month to thirty years with minor
gaps. In this market, it is clear that there is no scarcity of short-term data. In fact,
this is true especially due to the existence of a large number of Treasury bills issued with
maturities of four, thirteen, twenty-six, and fifty-two weeks.
4 Empirical applications
In this section, we provide two empirical applications for the VRP-SV model. The first
consists of modeling and fitting the Brazilian real and US nominal term structures of
interest rates. We compare in-sample and out-of-sample yield fitting results of the VRP-
SV model with those obtained by a fully parametric Svensson model and a pure spline-
based model (VRP model). In the second application, we use the time series of models’
parameters to estimate VaR measures for portfolios exposed to interest rate risks in the
Brazilian and US markets. We compare VaR accuracy obtained by the VRP-SV model
to those obtained by the aforementioned competing models.
Before describing our findings, we set up some features of the VRP-SV and VRP
models that are used in both applications. We first explain how we determine the thresh-
old maturity ¯τin which segmentation occurs in the VRP-SV model. Then, we define
the penalty function α(.) for the VRP-SV model (see Section 2.1) and show how it is
estimated. In a similar way, we end this section defining for the VRP model the penalty
function and showing how to estimate it. For each feature (segmentation threshold ¯τ,
or penalty function α(.)) we separately show how to estimate/calibrate it when dealing
with the Brazilian real market or the US Treasury market.
First, how should we choose the segmentation threshold ¯τ? We start trying to an-
swer this question by considering the Brazilian real market. Using the X12-ARIMA
methodology, we verify that the current Brazilian inflation time series presents seasonal
components only for terms below one year. Similarly, for predicted inflation, seasonal
components appear for terms below eleven months. In an effort to control these season-
ality effects when modeling the yield curve, we set the segmentation threshold at one
year (¯τ= 1). Therefore, the short end covers the interval [0,¯τ], while medium-term
and long-end segments cover (¯τ , 45]. Since, as explained in Section 2, in the VRP-SV
20
model the term structure is modeled as a spline in the short end, and as an approximated
Svensson model at longer-maturity segments, we have to choose spline knots too. For
maturities below one year (¯τ), we fix an ad-hoc set of spline knots at {0,0.25,0.5}, and
for maturities above one year, the knots used to approximate the Svensson model are
determined by a knot location algorithm (see the appendix).
For the US nominal case, we are interested in analyzing distortion effects in yield curve
estimation caused by a local short-term factor identified by Knez et al. (1994). This factor
mainly drives Treasury bill yields. Since the highest maturity for a Treasury bill is of
fifty-two weeks, in principle, we could fix ¯τagain at one year, exactly as in the Brazilian
real yield curve case. However, we explore a more general approach in determining the
segmentation threshold to verify how robust the VRP-SV model is to the choice of such
a threshold. We set ¯τto minimize the average out-of-sample model-implied yield errors
from a training sample. The training sample includes a panel of data in time series and
cross-sectional dimensions. In what follows, we describe how to measure model-implied
yield errors and how we choose the securities for the out-of-sample group of the training
sample.
Using the model-fitted price for each bond, we get its model-implied yield. The bond
yield error is defined as the absolute difference between model-implied yield and observed
yield. Following Bliss (1996) and Waggoner (1997), for each date of our analysis, we sort
the available securities by maturity and alternately include them into the in-sample and
out-of-sample groups. At each date, the first group is used to estimate the yield curve.
Then, this estimated yield curve is used to calculate the out-of-sample yield error for the
second group.
We test different values for ¯τand choose the one that minimizes the average out-of-
sample yield error, calculated using a training sample constituted by the first six years
of US monthly data (January 1995 to December 2000). More specifically, we test ¯τequal
to zero (no segmentation, i.e. Svensson model), one, two, three and four years. Seeking
to keep model parsimony, after choosing ¯τ, we allocate only three knots equally spaced
within the interval [0,¯τ], with the first knot equal to zero.
The average out-of-sample yield errors for the US training sample are 2.80, 2.56, 2.50,
2.51 and 2.60 bps, respectively for ¯τ= 0,1,2,3,4. Therefore, we set ¯τ=2. Nonetheless,
it should be clear that model performance is quite similar when segmentation occurs
21
between two and three years (¯τ= 2,3), slightly worse for ¯τ= 1,4, and the worst possible
for the no-segmentation case (¯τ= 0). The set of knots for the longer-maturity segment
τ , 30], is obtained via our knot-location algorithm described in the appendix.
Regarding the penalty function α(.) for the VRP-SV model, we specify it as a constant
over the entire maturity domain. A more complex structure for this function is not
necessary, since our model guarantees more smoothness in the long end even for a constant
penalty. Once again, we use minimization of the average out-of-sample yield errors to
choose the parameter αfrom a set of possible values given by gridα={0, 1e-8, 1e-7, ...,
1e8}. There is one specificity in the Brazilian market. Due to the limited number in the
cross-section of bonds, we avoid splitting them into in-sample and out-of-sample groups.
Instead, the out-of-sample error is calculated by using the “leave one out” procedure,
precisely as done by Anderson and Sleath (2001) for the UK curve.11 The training
sample for the Brazilian market uses the first two years of the daily data set described
in Section 3. For the Brazilian and US cases, the penalty parameter αvalues chosen are
1e-06 and 1, respectively.
In the VRP model, we adopt the three-tiered step function proposed by Waggoner
(1997) for the penalty function:
α(τ) =
s, if τ1
(s×l)0.5, if 1 < τ 10
l, if τ > 10
,
where sand lare parameters to be determined based on the out-of-sample technique
described above. The ad-hoc choice on steps’ locations represents the cutting points
for maturities defining Treasury bills, notes, and bonds. However, Waggoner (1997)
shows that, for the US market, this model’s fitting ability (both in- and out-of-sample)
is relatively robust to changes in steps’ locations and/or functional form for α(.).
For the Brazilian case, we adopt the same specification, since this functional form
is useful to capture volatility patterns observed for bond yields across a spectrum of
maturities. In the short end (τ < 1), due to seasonality effects, there are huge differences
11This consists of estimating the yield curve once for each bond in the cross-section, and in each
estimation, leaving out one specific bond and calculating the bond-yield error for this omitted security.
The model-implied out-of-sample yield error for any specific date is the average of all those obtained
bond-yield errors.
22
among yields, even for close maturities. In contrast, in the long end (τ > 10), yields
assume a similar asymptotic behavior. Finally, for 1 < τ < 10, there are moderate
variations across yields.12 For the Brazilian and US cases, the pairs of parameters (s,l)
estimated are respectively (1e-4, 1e+1) and (1e-1, 1e+4).
Last, when estimating the VRP model for the US nominal yield curve, we follow Fisher
et al. (1995) and Waggoner (1997), using a number of knots approximately equal to one
third of the number of securities used in the estimation process. Such knots are placed
approximately at the maturity of every third bond. In contrast, for the Brazilian real
term structure, we place a knot at the maturity of every bond used in the estimation,
since in this case we have a considerably smaller number of securities available in the
cross-section.
4.1 Fitting the real and nominal term structures
Brazilian real yield curve
To estimate the Brazilian real yield curve, we use daily data covering the period from
January 2009 to May 2014. Data consists of prices and characteristics of NTN-B bonds
coming from ANBIMA’s database, and prices of synthetic bonds with maturities of one
day, three months and six months. Here, we remind the reader of the importance of
including short-term synthetic reference bonds to identify the short end of the real yield
curve. We show below that failing to include these instruments (in a Svensson model)
distorts estimation of the one-year real yield, a fundamental instrument for monetary
policy analysis, in some periods across time.
Panel A of Table 1shows the average absolute in-sample yield error (in bps) for each
model. Estimation is based on the whole cross-section of bonds, and uses the whole period
of data (January 2009 to May 2014). SV refers to the Svensson model, without inclusion
of the three synthetic short-term reference bonds. In contrast, SV-Ref contemplates this
inclusion. Both VRP and VRP-SV models use these synthetic bonds in their estimation
procedure. However, for comparison purposes, we do not consider such bonds in the
calculations of yield errors, since the SV model is estimated without them.
12We have tested other step locations for the Brazilian case, such as {[0, 1] [1, 5] [5, 45]}and {[0, 5]
[5, 10] [10, 45]}. However, fitting results were inferior.
23
Table 1: Yields’ fitting errors for the Brazilian market.
<1y 1y-5y 5y-10y >10y All
Panel A: In-sample errors
SV 0,34 2,02 2,79 2,06 2,11
SV-Ref 80,11 15,24 7,67 6,18 12,16
VRP 14,98 4,44 1,72 1,24 2,88
VRP-SV 6,11 3,33 3,46 2,30 2,99
Panel B: Out-of-sample errors
SV 514,06 25,97 3,81 2,71 30,53
SV-Ref 134,43 19,88 7,21 6,75 16,08
VRP 31,48 10,88 3,50 2,53 6,41
VRP-SV 35,21 10,77 4,27 2,90 6,83
Notes: Reported statistics are the mean absolute errors in bond yields, and they are
expressed in basis points. Errors are calculated for the maturity ranges: less than
one year, one to five years, five to ten years, above ten years, and for all maturities.
The in-sample error for the SV model is, on average, about two bps, when we consider
all the maturities. This is a small value, since the average bid-ask spread is about eight
bps, and the average yield in this market is 5.4%. Note also that the error approaches
zero at the short end since the model has considerable freedom to price a small number
of bonds. In fact, in most cases, there is only one or no bonds with maturity less than
one year. However, by the fitting errors of the SV-Ref model, we observe that the
short end errors for the Svensson model increase dramatically when such a region is
populated, achieving unacceptably high values (on average, 80 bps). Moreover, when we
compare SV-Ref’s errors to SV’s errors, we observe that errors increase considerably for
the whole spectrum of maturities when synthetic bonds are included in the estimation
process. Therefore, this pervasive increase in fitting errors documents that the presence
of securities in the short end affects estimates of the whole yield curve in the Svensson
model.
The VRP model has an average in-sample error slightly larger than the SV model error
and, as expected, the inclusion of short-term securities (synthetic bonds) in the VRP
estimation process does not impact its goodness of fit for the medium-term and long-
end segments of the yield curve. Finally, the VRP-SV model has an average in-sample
error close to that of the VRP model, but errors are more homogeneously distributed
across maturities. The VRP-SV model is capable of fitting the whole yield curve in
a parsimonious way. Indeed, to describe the term structure above one year, both the
Svensson model and the VRP-SV use the same number of parameters.
24
jan/09 jan/10 jan/11 jan/12 jan/13 jan/14
0
2
4
6
8
10
annual rate %
Maturity: 6 months
jan/09 jan/10 jan/11 jan/12 jan/13 jan/14
0
2
4
6
8
annual rate %
Maturity: 1 year
jan/09 jan/10 jan/11 jan/12 jan/13 jan/14
2
3
4
5
6
7
8
annual rate %
Maturity: 5 years
jan/09 jan/10 jan/11 jan/12 jan/13 jan/14
4
5
6
7
8
annual rate %
Maturity: 20 years
SV SV-Ref VRP VRP-SV
Figure 5: Fitted Real Rates
The in-sample fitting accuracy makes us wonder why we should abandon the Svensson
model if it presents the best behavior. Below, we aim to convince the reader that whenever
a model avoids using short-term references, it generates completely unrealistic short-term
yields and sometimes even distorts yields of other maturities close to the short end. This
is exactly what happens with the Svensson model.
Figure 5compares model-implied zero-coupon yields for different maturities. Note
that the SV model presents completely unrealistic fitted yields for the short end, compro-
mising even the one-year rate at some points in time. In contrast, when for the six-month
yield, we compare estimates of the VRP and VRP-SV models to the synthetic six-month
yield (not plotted in the picture), we see that both models deliver yields extremely close
to the synthetic ones.
Despite the SV model’s unrealistic behavior for short-term yields, for the medium-
term and long-end segments, the estimates it provides are very similar to those provided
25
by the VRP and VRP-SV models. In contrast, attempting to fit the short-term references,
the SV-Ref model presents fitted short-term yields that are far from the corresponding
yields obtained by the VRP and VRP-SV models. In addition, it presents some less-
pronounced differences in larger-maturity yields appearing in the long end of the curve (5
and 20 years). Lastly, model-implied yields for the VRP and VRP-SV models are close
to each other across the whole range of maturities.
Now, Panel B of Figure 1reports the average out-of-sample absolute error of each
model. Errors are obtained according to the leave-one-out procedure described before.
The SV model presents, by far, the largest errors. Its performance is especially bad for
maturities below five years. Once again, it is possible to note that the scarcity of data
in the short end may compromise this model’s ability to fit yields above the one-year
term. As expected, the SV-Ref model also performs badly, since its entire fit is highly
affected by the short-term references. Even so, this model provides a better result, on
average, than the SV model. The VRP model has the smallest errors and it is only slightly
outperformed by the VRP-SV model within the range of one to five years. It is important
to emphasize that the out-of-sample performance of these two models is quite similar,
reproducing the same pattern obtained with the in-sample exercise. Therefore, it should
be clear at this point that the segmented VRP-SV model produces results comparable to
those obtained by the non-parsimonious VRP model.
US nominal yield curve
As discussed before, for each day in our database we split bonds into two groups (in-
sample and out-of-sample). The in-sample group is used to estimate the term structure
and calculate in-sample yield errors. This estimated yield curve is used to price out-of-
sample bonds and obtain the corresponding out-of-sample errors.
In this section, we separate our analysis into three different periods: January 1995 to
December 2007, January 2008 to December 2009, and January 2010 to December 2016.
We do that in order to isolate an anomaly in the Treasury market caused by extreme yield
discounts due to illiquidity among securities during the 2008-2009 crisis, as documented
by Gurkaynak and Wright (2012) and Musto et al. (2015). The latter authors show that
during the crisis, Treasury bonds, having cash flows exactly matched by corresponding
Treasury notes, were traded at a relative discount (with respect to notes) that reached
26
six percent of the bonds’ face value. The consequence of such an extreme violation of the
law of one price, which is assumed to be valid by all standard yield curve models, is the
burden of very large model-implied bond-yield errors.
Our main interest in this empirical illustration is in analyzing if the short-term local
factor identified by Knez et al. (1994) might cause a distortion in the estimation of the
whole term structure. A simple way to verify this conjecture is to perform an out-
of-sample yield error analysis to check if the medium-term and long-end segments are
negatively affected by the presence of short-term securities in the estimation process. To
this end, we estimate a Svensson model with all bonds in the in-sample group (SV) and
also using only bonds with maturities above one, two, and three years (SV1, SV2 and
SV3, respectively). After that, we compute the out-of-sample model-implied yield errors
for maturities in the range of five to ten years and above ten years, here representing,
respectively, the medium-term and long-end segments. These statistics appear in Table 2.
Observe that for the periods January 1995 - December 2007 and January 2010 - December
2016, out-of-sample errors for the medium-term and long-end segments decrease when
securities with maturities below two years are excluded from the estimation process.
Moreover, comparing SV1 and SV2 errors, we note that most of this decrease in errors
comes from an exclusion of Treasury Bills. Now that we have documented an existing
distortion in Svensson’s yield estimates due to the short-term Treasury bill factor, we
analyze whether the VRP-SV model is able to avoid such an increase in the out-of-sample
fitting errors.
Table 2: Short-term securities’ effect on the yield curve estimation
Model Jan 1995 - Dec 2007 Jan 2008 - Dec 2009 Jan 2010 - Dec 2016
5y-10y >10y 5y-10y >10y 5y-10y >10y
SV 4.16 1.96 11.18 6.50 2.17 2.67
SV1 3.31 1.15 11.47 5.21 1.76 1.86
SV2 2.81 1.12 11.81 4.24 1.64 1.74
SV3 2.89 1.06 9.48 4.68 1.60 1.59
Notes: Reported statistics are the mean absolute out-of-sample errors in bond yields, expressed in
basis point. “SV” refers to the Svensson model. “SV1/2/3” refers to the Svensson model estimated
using securities with maturity greater than 1/2/3 years. Errors are calculated for the maturity
ranges of five to ten years and above ten years.
Table 3presents in-sample and out-of-sample yield errors for different models and
periods. First, we compare VRP-SV out-of-sample errors with the corresponding errors
27
for the Svensson model presented in the previous table. For the periods January 1995 -
December 2007, and January 2010 - December 2016, the VRP-SV errors (for medium-term
and long-end segments) are similar in magnitude to the errors obtained by the Svensson
model estimates excluding Treasury bills (SV1 model). During the crisis years 2008-
2009, the VRP-SV model presents errors slightly higher than those for the SV1 model for
yields within the medium-term segment, but much lower errors for the long end of the
yield curve. Aggregating errors for the medium-term and long-end segments, the VRP-
SV model has an average error of 7.94 bps, while the SV1 model has a corresponding
8.19 bps average error. Most interestingly, the smaller error for the VRP-SV model is a
substantial result because it is estimated including Treasury bills, while the SV1 model
eliminates bills from the estimation process. Therefore, we identify that the proposed
VRP-SV model is able to avoid distortions in yield curve estimation that could be caused
by the short-term Treasury bill local factor documented by Knez et al. (1994) and Greg
(1996). In particular, the VRP-SV model includes short-term yields in estimation without
distorting yields for maturities in the medium-term and long-end segments.
Table 3: Fitting errors for the US market.
Maturity
Range
In-Sample Error Out-of-Sample Error
SV VRP-SV VRP SV VRP-SV VRP
Jan 1995 - Dec 2007
<1y 3.38 2.70 1.93 3.83 3.22 2.77
1y-5y 2.33 2.07 1.52 2.49 2.34 1.93
5y-10y 4.09 3.19 2.19 4.16 3.25 2.52
>10y 1.85 1.21 0.75 1.96 1.31 0.85
All 2.56 2.08 1.47 2.72 2.29 1.84
Jan 2008 - Dec 2009
<1y 4.61 2.79 1.88 4.43 3.04 2.57
1y-5y 4.68 4.10 2.39 4.79 4.55 3.12
5y-10y 10.43 10.49 8.78 11.18 11.71 11.42
>10y 5.53 4.21 2.58 6.50 4.57 3.03
All 5.86 5.04 3.49 6.22 5.59 4.50
Jan 2010 - Dec 2016
<1y 2.02 1.56 0.94 1.99 1.64 1.17
1y-5y 1.91 1.46 1.03 2.00 1.59 1.33
5y-10y 2.16 1.76 1.04 2.17 1.80 1.30
>10y 2.65 1.70 0.96 2.67 1.74 1.04
All 2.09 1.57 1.01 2.14 1.66 1.25
Notes: Reported statistics are the mean absolute errors in bond yields, expressed in
basis points. Errors are calculated for the maturity ranges less than one year, one to
five years, five to ten years, above ten years, and for all maturities.
28
Now, we focus in Table 3on comparing the Svensson, VRP, and VRP-SV models
using the whole range of maturities. A direct comparison between the SV and VRP-SV
models reveals that SV is outperformed both in-sample and out-of-sample. Moreover,
except for years 2008 and 2009, most of the VRP-SV gains against SV come from the
medium-term and long-end segments. We observe a decrease of almost one basis point
on average errors for the 5- to 10-year maturity range (January 1995 - December 2007)
and a similar decrease on average errors for the over-10-year maturity range (January
2010 - December 2016). As a consequence of that, the VRP-SV model presents a more
homogeneous error distribution within each maturity range than the SV model.
The VRP model’s in-sample errors are considerably lower than those of other models.
This is explained by its larger number of parameters: approximately 30, on average.13
When we change our perspective from the in-sample to the out-of-sample metric, the
increase in VRP average errors is, in general, larger than the corresponding increase for
the other models. Despite this fact, it still outperforms the SV and VRP-SV models in
terms of out-of-sample yield errors.
Taking a broader perspective, our proposed VRP-SV model accounts for approxi-
mately half of the fitting gains that the VRP model obtains, when compared to the SV
model. The VRP-SV gains come mainly from elimination of distortions in the para-
metric yield curve estimates for medium- and long-term yields, as noted before. Most
importantly, the VRP-SV model keeps a parsimonious number of parameters, an essen-
tial feature in several applications, including risk management, as we show in the next
section.
The gains achieved by VRP and VRP-SV over SV in this nominal yield curve illus-
tration are clearly lower than the gains obtained in the Brazilian government bond real
market. One obvious reason for such a difference is that inflation seasonality patterns
cause variations in the cross-section of short-term real yields that are hardly ever observed
in a nominal term structure. But a second, less obvious reason for observing smaller dif-
ferences in error fitting between parametric and spline models is that while the inflation
seasonality effect is always present, the dynamics of the local factor in the Treasury Bill
market may be less regular and change over time. If this is the case, calculating average
13We have, on average, 170 securities in each cross-section; we use half of them to take part in the
in-sample group. The number of knots chosen is approximately one third of the number of securities in
the in-sample group. With all these considerations, we use approximately 30 parameters for this model.
29
fitting errors for any extended period of time will not perfectly capture the local factor
time-varying effects on the distortion of curve estimation. To investigate this conjecture,
in Figure 6we plot the time series of the models’ out-of-sample errors for the medium-
term and long-end segments covering all sample years, excluding only 2008 and 2009 to
avoid a scale problem. The chosen maturity ranges are the ones for which the Svensson
model has its worst comparative performance.
1995 1999 2003 2007
0
2
4
6
8
basis point
Maturity range: above ten years
1995 1999 2003 2007
0
5
10
15
basis point
Maturity range: five to ten years
2010 2012 2014 2016
0
5
10
15
basis point
Maturity range: five to ten years
2010 2012 2014 2016
0
2
4
6
8
basis point
Maturity range: above ten years
SV VRP VRP-SV
Figure 6: Out-of-sample errors
We observe in Figure 6that the errors’ time series for the VRP and VRP-SV models
are quite similar, with the second constantly slightly higher than the first. For the SV
model, errors are higher than those from VRP and VRP-SV in general. Most impor-
tantly, during some periods or in isolated months, the SV errors diverge from the others,
presenting large differences of four or more bps. It is also possible to note that the long
end is more commonly affected than the medium-term segment.
These bond-yield error comparisons represent a useful way to detect distortions in
yield curve estimates, but do not show how these distortions really affect zero coupon
rates. Their effect should be much larger than the effect on the yield-to-maturity of a
bond. For instance, in 2016 we can observe that errors at the long end were larger for the
30
SV model than for the other two models. In July 2016, this difference between out-of-
sample errors for the SV model and the other models was around 2.30 bps. However, the
discrepancy in yield curve (meaning the term structure of interest rates) estimates is far
larger than that. Panel (a) of Figure 7shows estimated term structures of interest rates
for the SV, VRP and VRP-SV models. In panel (b) we report the difference between
the SV yield curve and the corresponding yield curves implied by the VRP, VRP-SV and
SV1 models. For the long end, it is possible to see that SV estimates diverge from the
other ones with a spread interval between -10 and 10 bps. For model-implied forward
rates, the difference range is even larger: between -80 and 20 bps. The corresponding
forward rate Figures are available upon request.
0 10 20 30
0
0.5
1
1.5
2
2.5
annual rate %
τ
ττ
τ
(a) Yield curve estimates
0 10 20 30
-15
-10
-5
0
5
10
15
basis point
τ
ττ
τ
(b) Deviation from SV estimate
SV VRP VRP-SV SV1
Figure 7: US yield curve estimation on July, 29, 2016
4.2 VaR
4.2.1 Methods and tests
We estimate a VaR model for portfolios composed of coupon-bearing bonds. Given
any portfolio and confidence level, the one-day VaR estimates the highest possible daily
portfolio loss. In other words, the time tVaR, with confidence level 1 - ϑ, is given by the
ϑ-quantile of the time tconditional distribution of one-day portfolio returns.14 Therefore,
in any VaR exercise, we are interested in forecasting the conditional distribution of returns
14That is, V aRϑ,t F1
t(ϑ), where Ft(.) is the conditional distribution of returns and F1
t(.) is its
inverse.
31
(Ft(.)), and more specifically, its lower quantiles.
Before getting to the estimation of the conditional distribution of returns, we must de-
termine which factors affect portfolio returns. Since a coupon-bearing bond has multiple
cash flows paid on different future dates, its returns depend on several interest rates with
different maturities. Thus, in general, portfolios of coupon-bearing bonds are exposed
to systematic interest rate risk (Knez et al. (1994)) as opposed to specific maturities
interest rate risks. This exposure to systematic interest rate risk implies that forecasting
Ft(.), in the end, boils down to forecasting the distribution of term structure movements.
Such movements are described by changes in the yield curve model’s vector of parameters
βt. Observe that parsimonious term structure models can map systematic interest rate
risks into a low-dimensional vector of parameters representing movements. This is a key
characteristic for the success of these models in producing computationally-feasible VaR
estimates.
We adopt two of the main techniques used to estimate fixed income portfolios’ VaR:
Historical Simulation (HS) and Monte Carlo Simulation (MCS). To introduce some no-
tation, assume that we want to estimate the VaR of a portfolio at date T, using a size J
window of past data. Using the HS technique, we generate J scenarios for the term struc-
ture by adding historical daily changes of yield curve factors (∆βt, t =TJ, ..., T 1)
to their current value βT. For each generated scenario, we use equation (2) to obtain the
model-implied prices (and returns) for each bond in our portfolio, and use these returns
to calculate the portfolio-implied return. The V aRϑis the ϑ-quantile of these Jportfolio
returns’ scenarios.
Using the MCS technique, we specify the dynamics for the vector of parameters βt
to follow the Vector Autoregressive Model of order one used by Diebold and Li (2006):
βt+1 = Γβt+t+1, with t+1 N (0,Σ). We draw a sample for t+1 generating Jscenarios
for βt+1. As in the HS case, in each scenario, we use equation (2) to obtain the portfolio
return. And, again, the V aRϑ,t+1 is the ϑ-quantile of these Jportfolio returns’ scenarios.
Time-varying volatility is one of the most important stylized facts in returns data (see
Engle (1982)). So, to avoid poor results in our VaR estimation, we want to accommodate
this feature in our model. Building on the work of Hull and White (1998) and Duffie
and Pan (1997), instead of keeping Σfixed in the MC simulation, we use a Generalized
Autoregressive Conditional Heteroskedastic process (GARCH, Bollerslev (1986)). This
32
introduces a time-varying volatility component on the term structure movements (∆βt+1).
We use time tinformation in the GARCH process to forecast the next period’s variance
matrix ˆ
Σt+1. This matrix is directly plugged into the MCS process to obtain a V aRϑ,t+1
estimate.
To introduce time-varying volatility to the HS technique, it takes slightly more work.
According to Hull and White (1998), each ∆βt, with t(TJ,TJ+ 1, ..., T2,
T1), should be standardized using the GARCH process:
βt,std = (Σ1/2
tΣ1/2
T)∆βt,(20)
where Σtis the ∆βtGARCH variance, estimated based on the rolling window [tJ, t1].
The result is a sample of simulated standardized daily changes for the yield curve parame-
ters, IT={βk,std}T1
k=TJ. For robustness against outliers, we add a bootstrapping refine-
ment to the HS method, generating Mbootstrap samples, {I(1)
T, ..., I(M)
T}, where for each
i= 1, ..., M ,I(i)
Thas size J, and is obtained by sampling from ITwith replacement ∆βt,std.
For each I(i)
T, we obtain a sample of scenarios for the portfolio returns and a corresponding
ϑ-quantile, V aR(i)
ϑ,t+1. The final VaR is given by: V aRϑ,t+1 =PM
i=1 V aR(i)
ϑ,t+1/M.15
It is only feasible to use a combination of MCS/HS with time-varying volatility
(GARCH) when we adopt a parsimonious model for the yield curve. Just to illustrate this
fact, if we choose the VRP model to estimate the term structure of interest rates using the
whole set of available bonds in the Brazilian (or US) market, the model will have seven-
teen (or fifty nine) parameters to fit the yield curve. This implies that in order to obtain
the historical covariance matrix Σ, this model would demand an estimation of around
150 (or 1770) additional covariance parameters. It would be too costly to model the time-
varying dynamics of such covariances using a traditional GARCH model. Therefore, the
choice of estimating fixed income portfolios’ VaR using the VRP model with MCS/HS
is only computationally acceptable without combining it with time-varying volatility for
yield curve movements. This compromises the ability of this model to produce adequate
VaR estimates.
To measure the adequacy of estimates for a given set of percentiles ϑgenerated by
the VaR techniques we just described, we implement tests of unconditional coverage
15See Dowd (2005) for additional details on the use of bootstrapping procedures to estimate robust
VaR. In our bootstrap refinement robust to outliers, we set M= 10000.
33
(UC), independence (IND) and conditional coverage (CC), as proposed by Christoffersen
(1998).16 The null hypotheses (H0) for the UC test states that the exception rate is equal
to ϑ. The H0 for the IND test states that all VaR violations are independently distributed
over time. And, the H0 for the CC test states that VaR violations are independently
distributed and that the probability that one exception happens is equal to ϑ. In addition
to tests for specific percentiles, we also implement the Kuiper test (K), which focuses on
the entire probability distribution Ft(.), as shown by Crnkovic and Drachman (1996). Its
H0 states that the predicted distribution of variations in portfolio values is equal to the
realized distribution. In our analysis, a VaR model is considered well-adjusted when none
of the null hypotheses of the tests proposed above are rejected. There can be multiple
well-adjusted VaR models. Also, given two well-adjusted models, we cannot determine
which is best based on criteria used to measure VaR adequacy.
In cases of multiple well-adjusted models, we can evaluate their performance and
rank them by measuring the error in their predicted ϑ-quantiles. This metric is easily
motivated by regulatory and trading purposes, since when a VaR violation occurs in a
certain model, it is desirable that the VaR estimate and the realized loss are as close
as possible to each other. Following Giacomini and Komunjer (2005) and Engle and
Manganelli (2004), the quantile predictive error can be measured by the loss function
used in quantile regressions L(ϑ, t)=(ϑ(rtV aRϑ,t <0))[rtV aRϑ,t ], where rt
is the daily portfolio return and (.) is an indicator function that assumes one when a
violation occurs.
In the context of ranking well-adjusted models, to evaluate if the performance of two
models are statically different, we implement the Giacomini and White (2006) test, in
both its unconditional and conditional versions. The unconditional version coincides with
the Diebold and Mariano (1995) test and its H0 states that on average the two models
have equal performance. If H0 is rejected, we select the model that has lower L(.). The
conditional version introduces a vector of t-measurable instruments ht. Its H0 states
that on average the two models have equal performance at time t+ 1, conditional on
the information provided by the instruments ht. We follow Giacomini and Komunjer
(2005) and use as instruments a constant, the current portfolio return, and a measure of
16The exception rate (percent of VaR violations) for V aRϑof a certain portfolio is, by definition,
the proportion of time that the portfolio loss is larger than the estimated V aRϑ. For unbiased VaR
estimates, we should expect the exception rate to be close to ϑ.
34
current portfolio return volatility, estimated based on the previous two weeks. When H0
is rejected it means that hthas useful information to predict the difference in models’
performance at time t+ 1. Thus, for each day of our analysis, we calculate the difference
in conditional performance one day ahead (that is, the time t+1 difference in performance
conditional on ht), and select the model that has lower predicted L(ϑ, t + 1). A model
has better conditional performance if it is selected on more than 50% of the days.
4.2.2 Application and results
We build several portfolios containing one unit of each bond available within different
ranges of maturities. In our analysis of the Brazilian real market, we consider three
portfolios: BR General, BR Short and BR Long. The first, BR General, uses the entire
range of maturities. The BR Short portfolio works with maturities lower than five years,
and the BR Long covers the range of maturities above five years. For the US case,
since there is a much larger number of bonds, we include an Intermediate portfolio. The
US General portfolio has the same definition as BR General. The US Short portfolio
considers maturities below two years, while the US Intermediate portfolio works with a
range of maturities between two and seven years. Finally, the US Long portfolio contains
bonds with maturities above seven years.
To estimate VaR, we use a rolling window containing 504 daily observations, covering
data from two years of business days. We split the available data sets into two parts. The
first, containing 504 days, is used to initialize our VaR estimates. For the second part,
which comprises data on 828 subsequent days for the Brazilian data set and 1248 days
for the US data set, we compute VaR estimates for each day. The first part of the data
set is also used to determine λ1and λ2in the VRP-SV and Svensson models, based on
a minimization of the global in-sample model fitting error. Taking those parameters as
fixed across time, we make the functional basis of these models also constant over time,
guaranteeing temporal consistency for the models’ parameters βt.
For the VRP-SV and Svensson models, we produce VaR estimates using the HS and
MCS methods with and without a GARCH process for the variance of term structure
movements. Considering the Brazilian and US cases and both yield curve models (VRP-
SV and Svensson), the HS method with time-varying GARCH volatility provides the best
adjusted VaR estimates, when compared to alternative VaR models such as HS without
35
Table 4: VaR results for the Brazilian market
General Portfolio Short Portfolio Long Portfolio
Model ϑ
Exception
rate IND UC CC K
Exception
rate IND UC CC K
Exception
rate IND UC CC K
SV
5,0% 5.0% 0.02 0.95 0.05
0.01
3.5% 0.37 0.04 0.08
0.00
5.8% 0.02 0.30 0.04
2.5% 2.5% 0.30 0.95 0.58 1.3% 0.59 0.02 0.05 2.7% 0.27 0.77 0.53 0.00
1.0% 1.4% 0.55 0.22 0.40 0.5% 0.84 0.10 0.25 1.4% 0.55 0.22 0.40
0.5% 0.6% 0.81 0.68 0.89 0.5% 0.84 0.94 0.98 0.6% 0.81 0.68 0.89
SV-Ref
5.0% 1.3% 0.59 0.00 0.00
0.00
0.2% 0.92 0.00 0.00
0.00
1.6% 0.52 0.00 0.00
0.00
2.5% 0.7% 0.77 0.00 0.00 0.2% 0.92 0.00 0.00 1.1% 0.66 0.00 0.01
1.0% 0.2% 0.92 0.01 0.03 0.1% 0.96 0.00 0.01 0.5% 0.84 0.10 0.25
0.5% 0.2% 0.92 0.24 0.50 0.0% 0.00 0.2% 0.92 0.24 0.50
VRP
5.0% 6.9% 0.00 0.02 0.00
0.00
6.6% 0.09 0.04 0.03
0.00
7.7% 0.00 0.00 0.00
0.00
2.5% 4.1% 0.00 0.01 0.00 3.5% 0.02 0.08 0.01 3.6% 0.00 0.05 0.00
1.0% 1.3% 0.13 0.37 0.21 1.6% 0.19 0.13 0.14 1.2% 0.10 0.56 0.23
0.5% 1.0% 0.06 0.09 0.04 1.1% 0.66 0.04 0.11 0.8% 0.04 0.20 0.06
VRP-SV
5.0% 4.7% 0.90 0.70 0.92
0.42
4.2% 0.66 0.30 0.53
0.48
5.1% 0.92 0.92 0.99
0.86
2.5% 2.3% 0.34 0.70 0.59 1.9% 0.43 0.28 0.40 2.4% 0.50 0.88 0.79
1.0% 1.2% 0.62 0.56 0.75 1.0% 0.69 0.92 0.92 1.2% 0.62 0.56 0.75
0.5% 0.6% 0.81 0.68 0.89 0.7% 0.77 0.39 0.66 0.6% 0.81 0.68 0.89
Notes: The exception rate and the p-values are for the tests of unconditional coverage (UC), independence (IND), conditional coverage (CC), and Kuiper (K).
The p-value in bold indicates that it is not possible to reject the null hypothesis. General Portfolio contains one unit of each available bond. Short Portfolio is
built by bonds with maturities lower than five years while Long Portfolio covers bonds with maturities over five years.
36
Table 5: VaR results for the US market
ϑModel
Exception
rate IND UC CC K Model
Exception
rate IND UC CC K Model
Exception
rate IND UC CC K
Panel A: General Portfolio
5.0%
SV
4.7% 0.60 0.66 0.79
0.63 VRP
4.4% 0.77 0.33 0.59
0.01 VRP-SV
5.0% 0.32 0.94 0.61
0.52
2.5% 2.3% 0.70 0.69 0.86 1.9% 0.33 0.17 0.25 2.2% 0.26 0.55 0.44
1.0% 0.9% 0.66 0.67 0.83 0.9% 0.66 0.67 0.83 0.9% 0.66 0.67 0.83
0.5% 0.6% 0.75 0.50 0.76 0.3% 0.87 0.34 0.62 0.6% 0.78 0.76 0.92
Panel B: Short Portfolio
5.0%
SV
4.6% 0.80 0.48 0.75
0.05 VRP
5.6% 0.97 0.33 0.62
0.48 VRP-SV
4.6% 0.80 0.48 0.75
0.18
2.5% 2.4% 0.75 0.83 0.93 3.3% 0.75 0.09 0.23 2.5% 0.80 0.97 0.97
1.0% 0.8% 0.69 0.46 0.71 1.3% 0.52 0.34 0.51 0.7% 0.72 0.30 0.54
0.5% 0.5% 0.81 0.92 0.97 0.8% 0.69 0.17 0.35 0.4% 0.84 0.61 0.86
Panel C: Median Porfolio
5.0%
SV
4.81% 0.18 0.75 0.39
1.00 VRP
4.6% 0.40 0.48 0.54
0.16 VRP-SV
4.6% 0.22 0.56 0.40
1.00
2.5% 2.40% 0.22 0.83 0.47 2.2% 0.26 0.55 0.44 2.6% 0.18 0.75 0.39
1.0% 1.20% 0.55 0.49 0.65 1.0% 0.63 0.89 0.88 0.8% 0.69 0.46 0.71
0.5% 0.40% 0.84 0.61 0.86 0.6% 0.78 0.76 0.92 0.6% 0.75 0.50 0.76
Panel D: Long Porfolio
5.0%
SV
4.9% 0.26 0.86 0.52
0.48 VRP
4.1% 0.95 0.13 0.31
0.00 VRP-SV
5.1% 0.35 0.84 0.63
0.60
2.5% 2.0% 0.31 0.24 0.30 1.7% 0.36 0.05 0.10 2.2% 0.26 0.55 0.44
1.0% 0.9% 0.66 0.67 0.83 0.7% 0.72 0.30 0.54 0.9% 0.66 0.67 0.83
0.5% 0.6% 0.75 0.50 0.76 0.2% 0.90 0.15 0.35 0.6% 0.78 0.76 0.92
Notes: The exception rate and the p-values are for the tests of unconditional coverage (UC), independence (IND), conditional coverage (CC), and Kuiper (K).
The p-value in bold indicates that it is not possible to reject the null hypothesis. General Portfolio contains one unit of each available bond. Short Portfolio is
built by securities with maturities lower than two years. Median Portfolio covers securities with maturities between two and seven years. Long Portfolio works
with securities over ten years.
37
GARCH, and MCS with and without GARCH. Therefore, we will focus on the results of
this method. For the VRP yield curve model, which has a large number of parameters,
VaR estimates can be obtained only via HS without time-varying GARCH volatility, due
to the otherwise high computational costs.
Starting with an analysis of the Brazilian real term structure, Table 4presents excep-
tion rates and p-values for UC, IND, CC and K tests, considering the quantiles usually
adopted by regulators and market practitioners. Considering all portfolios, quantiles and
return distributions, we reject almost all H0s for the VRP model. This indicates that the
ordinary HS method (without GARCH) cannot produce well-adjusted VaR estimates in
highly volatile markets. As we discussed earlier, for the Svensson model, there is the op-
tion of using short-term references or not. When we use short-term references (SV-Ref),
the results are even worse than those obtained by the VRP model. Once again, most null
hypotheses (H0s) of the VaR adequacy tests are rejected. In addition, exception rates
are now even more distant from their correspondent quantiles, than was the case with
the VRP model. When we estimate the Svensson model without references in the short
end (SV), we get better results. However, the H0s for all different tests continue to be
rejected for all portfolio types. In particular, for the BR General and BR Long portfolios,
we can reject H0s for ϑ=5% and for the distribution test K. For the BR Short portfolio,
the non-realistic volatility in the modeled short-term yields (due to a lack of short-term
references) has strong impact on VaR estimates. As a consequence, the model is rejected
at all quantiles except at ϑ=0.5%. In contrast, for the proposed VRP-SV model, we
cannot reject any H0, for any portfolio. Moreover, VRP-SV is the only model that can
correctly predict the conditional distribution function of portfolio returns, as indicated
by the distribution test K.
Table 5presents VaR quantile and distribution tests for the US nominal yield curve.
VaR estimates using the VRP model show better adequacy in comparison to those ob-
tained for the Brazilian real yield curve. With the quantile tests (UC, IND, CC), we only
reject H0 for the US Short and US Long portfolios. This indicates that a heteroskedastic
treatment for the volatility of term structure movements is important, but less essential
in this case. This might be a consequence of the stability of the term structure after 2010,
a period in which interest rates reached the zero lower bound. Note that we also reject
the H0 of the K distribution test for the US General and US Long portfolios. With the
38
SV model, we cannot reject H0 for all quantile tests. However, we reject H0 for the K
distribution test for the US Short portfolio. Once more, the proposed VRP-SV model is
the only method for which all H0s cannot be rejected considering any type of portfolio.
Focusing solely on the quantile tests, both VRP-SV and SV models offer well-adjusted
VaR estimates for all portfolios. Therefore, to select the best model of these two, we
should measure the performance of their predictive quantile errors.
Table 6: Tests for VaR estimates performance in the US market
VRP-SV against SV VRP-SV against VRP
ϑ= 0.5% ϑ= 1.0% ϑ= 2.5% ϑ= 5.0% ϑ= 0.5% ϑ= 1.0% ϑ= 2.5% ϑ= 5.0%
Panel A: General Portfolio
Unconditional
test
0.92 0.95 0.97 0.98 1.10 0.96 0.95 0.97
(0.00) (0.00) (0.03) (0.04) (0.24) (0.31) (0.09) (0.09)
Conditional
test
0.00% 0.00% 0.32% 1.68% 80.45% 0.08% 9.94% 5.29%
(0.00) (0.00) (0.10) (0.05) (0.35) (0.16) (0.20) (0.16)
Panel B: Short Portfolio
Unconditional
test
0.91 0.90 0.92 0.93 0.95 0.90 0.91 0.93
(0.12) (0.00) (0.00) (0.00) (0.66) (0.09) (0.00) (0.00)
Conditional
test
15.54% 1.76% 1.36% 3.37% 25.80% 14.66% 11.30% 10.26%
(0.01) (0.02) (0.02) (0.00) (0.00) (0.00) (0.00) (0.00)
Panel C: Median Portfolio
Unconditional
test
1.14 0.96 0.98 0.97 1.05 0.94 0.95 0.96
(0.15) (0.13) (0.16) (0.01) (0.47) (0.11) (0.04) (0.03)
Conditional
test
94.71% 0.00% 15.22% 4.81% 91.11% 0.00% 7.05% 14.18%
(0.43) (0.01) (0.28) (0.01) (0.57) (0.24) (0.12) (0.03)
Panel D: Long Portfolio
Unconditional
test
0.91 0.95 0.97 0.99 1.01 0.96 0.97 0.98
(0.04) (0.02) (0.04) (0.36) (0.91) (0.25) (0.17) (0.25)
Conditional
test
0.00% 0.00% 0.00% 14.74% 44.23% 20.59% 16.35% 19.63%
(0.10) (0.01) (0.01) (0.42) (0.68) (0.04) (0.06) (0.34)
Notes: For the unconditional test, the entries are the ratios of the average loss function for the
VRP-SV model to the competing one. Right below those entries, the numbers within parentheses are
the p-values of the test. For the conditional test, the entries are the proportion of times the competing
model outperforms the VRP-SV method according to the decision rule. The number within
parentheses are the p-values of the test.For both tests, the entries in bold indicate that the test rejects
equal predictive ability at the 10% level.
Table 6reports comparisons in performance between the VRP-SV model and the
SV and VRP models. For the unconditional test, we report a ratio between the average
value of the VRP-SV model’s loss function and the loss function of the competing method.
This means that ratios less than one indicate that, on average, the VRP-SV model has
a lower value for the loss function. For the conditional test, we report the proportion
39
of times in which the competing model outperforms the VRP-SV method. Thus, values
less than 50% indicate that the VRP-SV model has a better conditional performance.
The numbers within parentheses are the p-values of the respective tests. The entries in
bold denote that the test rejects equal performance ability at the 10% level. We can see
that the proposed VRP-SV model outperforms both the SV and the VRP models in all
portfolios, for different quantile values. As we would expect, gains in performance are
more pronounced for the US Short portfolio.
5 Conclusion
In fixed-income markets, the existence of segmentation between short-maturity and longer-
maturity instruments represents a challenge to term structure models aiming to estimate
an unbiased yield curve. Trying to circumvent this issue, we rely on both the flexibility
of spline models and the parsimoniousness of parametric models, to propose a segmented
term structure model (VRP-SV). VRP-SV combines these two classes of functions to
obtain the best of both worlds: flexible enough at the short end to fit the yield curve
without distorting it, and parsimonious at longer-maturity segments allowing for dynamic
empirical applications like hedging and risk management procedures with relatively low
computational costs. We consider splines to fit the short end while the medium-term and
long-end segments are captured by a parametric Svensson model.
We illustrate the benefits of VRP-SV in two empirical applications (curve fitting
and risk management), considering two very distinct markets: the Brazilian government
index-linked bond market, and the US Treasury market.
The first application consists of analyzing the out-of-sample ability in fitting the
Brazilian real yield curve, and the US Treasury nominal yield curve of three inherently
distinct classes of models: the VRP class (nonparametric spline model), the Svensson class
(parametric) and our proposed VRP-SV class (mixed model). Since both yield curves
present segmentation between their short end and longer-maturity segments, Svensson’s
long-term yields estimates end up being contaminated by the segmentation effect. As a
consequence, the VRP-SV and VRP models outperform the Svensson model, a benchmark
adopted by central banks.
In the second application, we estimate VaR measures for portfolios exposed to interest
40
rate risk. We find that VRP-SV consistently outperforms the Svensson and VRP models
in different tests of VaR adequacy, and in direct comparisons of the predictive ability to
forecast quantiles of the distribution of portfolios’ returns.
The VRP-SV model introduces a unified technique useful for pricing, hedging, risk
management, and monetary policy analysis.
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44
A Appendix
A.1 Local approximation
Let d= (dk)κ+p+1
k=1 be the set of knot points with d1=aedk+p+1 =b. Define Sp,dto be
the space of all linear combinations of B-splines with degree pand knot vector d.Lyche
and Morken (2011) show17 that for a function fCp+1
[a, b]18, the distance between f
and Sp,don the interval [dk, dk+1] is bounded by
inf
gSp,d
||fg||,[dk,dk+1]Kphp+1||Dp+1f||,[dkp+1 ,dk+p],(21)
where Dkfis the kth derivative of f,||.||,jis the maximum norm on interval j,h=
dkp+1 dk+p, and the constant Kponly depends on p, and is given by
Kp=(2p(p1))p+p!
2p+1(p+ 1)!p!.
By (21), with a sufficient number of knots, it is possible to make hsufficiently small,
such that the local error satisfies an acceptable tolerance level. Moreover, (21) also
informs that if we want to obtain homogeneous approximation errors across all regions,
we need to allocate more knot points in the regions where |Dp+1f|assumes larger values.
Therefore, the bound given by (21) not only ensures that the approximation is possible,
but also provides useful information on how to efficiently allocate the knot points.
A.2 Knot location algorithm
Our knot location algorithm is a forward incremental method using a heuristic rule that
comes directly from the local bound (21). Precisely, for a given λ, consider the functions
¯
ψk(.), with k∈ {1,2,3,4}and domain (¯τ , T ], as defined in (11). In this interval, allocate
a reduced number of knot points equally spaced, with the first one allocated very close
to ¯τand the last one at T.19 Then, letting dbe the set of those knots, for each interval
17See Corollary 9.12, Theorem 9.9 and Theorem 9.1
18A function fCk
[a, b], if it has k1 continuous derivatives on the interval[a, b], and the kth
derivative Dkfis continuous everywhere except for a finite number of points in the interior (a, b), given
by ∆ = (zj). At the points of discontinuity ∆ the limits from the left and right, given by Dkf(zj+) and
Dkf(zj), should exist so all the jumps are finite.
19On our applications, we make the first knot above ¯τto be ¯τ+ 0.00794 for the Brazilian curve and to
be ¯τ+ 0.00548 for the US curve. Those increments represent two business days expressed in years (2/252)
and two actual days expressed in years (2/360), respectively. In addition, when fitting the Brazilian and
45
=
7=7=+1 7=+2 7=+3 7=+4 7=+5
0
0.5
1
1.5
2
2.5
3
3.5
6=0.5
6=1
6=1.5
6=2
=
7=7=+1 7=+2 7=+3 7=+4 7=+5
0
2
4
6
8
10
12
14
6=0.5
6=1
6=1.5
6=2
(a) 4¯
ψ2
∂τ 4
=
7=7=+1 7=+2 7=+3 7=+4 7=+5
0
0.5
1
1.5
2
2.5
3
3.5
6=0.5
6=1
6=1.5
6=2
=
7=7=+1 7=+2 7=+3 7=+4 7=+5
0
2
4
6
8
10
12
14
6=0.5
6=1
6=1.5
6=2
(b) 4¯
ψ3
∂τ 4
Figure 8: Fourth derivative of Svensson functions
[dk2, dk+3], starting from the smallest knots to the largest ones, calculate
Supk=max
j∈{2,3}K3h4||D4¯
ψj||,[dk2,dk+3].(22)
If Supk> tol, where tol is the acceptable tolerance, we calculate a new knot point
d= (dk+3 +dk2)/2. Then, we update the set of knots dby adding d, and calculate
Supkonce more time. We continue the iteration until Supktol. When the condition
Supktol is achieved, we go to the next interval, updating k=k+ 1, calculate Supk,
and restart the process. The whole process finishes when all intervals [dk2, dk+3] in d
are covered.
Although it is possible to show that, τ > ¯τand λ > 0, |4¯
ψ2/∂τ 4|<0 and
|4¯
ψ2/∂λ4|>0, the same is not true for ¯
ψ3. Figure 8reports the absolute values for the
fourth derivatives of ¯
ψ2and ¯
ψ3. For those, we see that the largest values are found when
τapproaches ¯τ, and such values are even larger when λincreases. Thus, according to
(21), the number of necessary knot points is an increasing function of λ. To avoid the
necessity of calculating the vector of knots and corresponding B-splines in each iteration
when optimizing λ, we determine the set of knots that satisfies (21) for all possible
predefined λ’s. To do so, we start defining a grid with possibles values for λ, given by
gridλ= [0.2,0.25,0.3, ...5.90,5.95,6.0].20 After that, for each λjgridλ, we obtain the
US yield curve, we start with knot points equally spaced at every four years.
20In a similar spirit to Diebold and Li (2006), the grid range is determined based in the maturity where
46
2345678910
0
10
20
30
40
τ
Number of knots
Figure 9: Knots allocation
knot vector indexing it by dj. The final knot vector dis obtained as the union of all sets
dj, after elimination of any repeated sets.
Since Supkis an upper bound for the approximation error, the bound might be well
above the error, which is defined as
Error =max
j∈{2,3}|¯
ψj˜
ψj|.(23)
Therefore, creating a knot allocation algorithm that makes (21) always below a desired
tolerance level, automatically makes the approximation error below that tolerance level,
potentially at the cost of an allocation of more knot points than necessary. To control for
an unnecessarily large number of knots, we create a grid of possible values for tol, given
by gridtol ={1e3, 1e2,..., 1e-5, 1e-6}, and for each grid entry, we perform the knot points
allocation algorithm, and calculate the approximation error (23). We select the larger
value in gridtol and its associated knot vector dthat makes the approximation error in
(23) below a predefined tolerance level, which we take as 1e-6. When we perform such
an algorithm, the value in gridtol chosen is 1e-2.
Once we obtain the vector of knot points dabove ¯τ, we insert the knots below ¯τinto
that vector, completing the set of knots for the entire domain.
Figure 9shows the histogram of the set of knot points above ¯τfor the numerical
example of section 2.5, in which ¯τ= 2 and T= 10. As expected, the knots are more
concentrated around maturities near ¯τ.
the curvature function ¯
ψ3assumes its maximum. With λ= 6, ¯
ψ3reaches its maximum at τ= ¯τ+ 0.3,
a value very close to ¯τ. For this reason, ¯
ψ3is also very close to ¯
ψ2for λlarger than six.
47
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This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.
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This article gives a broad and accessible overview of models of value at risk (WR), a popular measure of the market risk of a financial firm's "book," the list ofpositions in various instruments that expose the firm to financial risk. Roughly speaking, the value at risk o f a portfolio is the loss in market value over a given time period, such as one day or two weeks, that is exceeded with a small probability, such as 1%. We focus narrowly on the market risk associated with changes in the prices or rates of underlying traded instruments. Traditionally, this would include such aspects of credit risk as the risk of changes in the spreads ofpublicly traded corporate and sovereign bonds. In order to maintain a narrow focus, however, I/aR does not traditionally include, and we do not review here, the risk of defdult on long-term derivative contracts. We begin with models of the distribution of underlying market returns and of volatility, emphasizing the roles of price jumps and of stochastic volatility in determining the "fatness" of the tails of the distributions of returns in various markets. We then turn to methods for approximating the value at risk of derivatives, using numerical approximations based on delta and gamma. Estimation of the value at risk o f a portfolio of positions is then discussed, beginning with the choice of which risk factors to include. An extensive numerical example illustrates the accuracy of various alternative methods, allowing for correlated jumps in the underlying market prices. Finally, we review "scenario analysis," which treats the potential losses associated with particular scenarios, such as a given parallel shift of a yield curve or a given change in volatility.
Book
Understanding the dynamic evolution of the yield curve is critical to many financial tasks, including pricing financial assets and their derivatives, managing financial risk, allocating portfolios, structuring fiscal debt, conducting monetary policy, and valuing capital goods. Unfortunately, most yield curve models tend to be theoretically rigorous but empirically disappointing, or empirically successful but theoretically lacking. In this book, Francis Diebold and Glenn Rudebusch propose two extensions of the classic yield curve model of Nelson and Siegel that are both theoretically rigorous and empirically successful. The first extension is the dynamic Nelson-Siegel model (DNS), while the second takes this dynamic version and makes it arbitrage-free (AFNS). Diebold and Rudebusch show how these two models are just slightly different implementations of a single unified approach to dynamic yield curve modeling and forecasting. They emphasize both descriptive and efficient-markets aspects, they pay special attention to the links between the yield curve and macroeconomic fundamentals, and they show why DNS and AFNS are likely to remain of lasting appeal even as alternative arbitrage-free models are developed. Based on the Econometric and Tinbergen Institutes Lectures, Yield Curve Modeling and Forecasting contains essential tools with enhanced utility for academics, central banks, governments, and industry.