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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 pi−πi(β(α))

wi2

+ZT

0

α(τ)h00(τ)2dτ !,(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)2dτ =β|ZT

0

α(τ)δ00(τ)|δ00(τ)dτβ=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 cije−yiτij. 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: pi−πi≈ −wi(yi−yπ

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)dτ. Using

the last equation, we can rewrite the minimization problem (3), for a given α, as follows:

min

β(α)P−Π(β(α))|WP−Π(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|(β−β0),(5)

where ∂Πi(β)

∂β|=−πi(β)ciDiδ(τi). Deﬁning X(β0) = ∂Π(β)

∂β|β=β0and Y(β0) = P−Π(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+

K−1

X

k=1

ηk|τ−sk|3,

for some constants {θ}p

j=1 and {η}K−1

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 ≤k≤K. A B-spline of degree pis deﬁned by

the following recursion:

φp

k(τ) = φp−1

k(τ−dk)

(dk+p−dk)+φp−1

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 dk≤τ < 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 βk∈R

∀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¯

k−3, φ¯

k−2, ..., 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(τ)Γk+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(τ)Γ1. 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)Γ.(12)

Given (10) and (12), the mixed basis Ψ(.), deﬁned over the whole domain [0, T ], is given

by

Ψ(τ) = (φ1(τ), ..., φ¯

k−4(τ))1×(¯

k−4) ˆ

ψ(τ)

=(φ1(τ), ..., φ¯

k−4(τ))1×(¯

k−4) ¯

φ(τ)

I(¯

k−4) 0(¯

k−4)×4

0¯κ×(¯

k−4) Γ

κ×¯

k

=φ(τ)A, (13)

where Ijis the identity matrix with dimension j, 0i×jis a i×jnull matrix, and

A=

I(¯

k−4) 0(¯

k−4)×4

0¯κ×(¯

k−4) Γ

κ×¯

k

.

The Ψ basis has dimension ¯

k, with four functions used to approximate the Svensson basis,

and ¯

k−4 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)AβM(15)

=πSp

i(βSp =AβM),(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 =AβM. 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(¯

k−4) 0(¯

k−4)×¯κ

0¯κ×(¯

k−4) 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 ¯

k−4 elements of

the mixed basis are free B-splines, Mdoes not restrict the ﬁrst ¯

k−4 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 ≤k≤11. 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 ¯

k−4, 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¯

k−4(τ))1×(¯

k−4) ¯

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.

0246810

5

10

15

20

τ

%

(a) Model VRP-SV

α=0 α=1 α=1e+2 α=1e+8

0 1 1e+2 1e+8

1

2

3

4

5

6

α

Number of parameters

0 1 1e+2 1e+8

0

0.5

1

1.5

2

2.5

x 10

-8

Approximation error

(c) Statistics VRP-SV

0246810

5

10

15

20 (b) Model Cubic Spline

%

τ

0 1 1e+2 1e+8

2

4

6

8

10

12

14

α

Number of parameters

(d) Statistics Cubic Spline

Number of parameters Approximation error

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 τ≤1

(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 ≡F−1

t(ϑ), where Ft(.) is the conditional distribution of returns and F−1

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 =T−J, ..., 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∈(T−J,T−J+ 1, ..., T−2,

T−1), should be standardized using the GARCH process:

∆βt,std = (Σ−1/2

tΣ1/2

T)∆βt,(20)

where Σtis the ∆βtGARCH variance, estimated based on the rolling window [t−J, t−1].

The result is a sample of simulated standardized daily changes for the yield curve parame-

ters, IT={∆βk,std}T−1

k=T−J. 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−(rt−V aRϑ,t <0))[rt−V 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 f∈Cp+1

∆[a, b]18, the distance between f

and Sp,don the interval [dk, dk+1] is bounded by

inf

g∈Sp,d

||f−g||∞,[dk,dk+1]≤Kphp+1||Dp+1f||∞,[dk−p+1 ,dk+p],(21)

where Dkfis the kth derivative of f,||.||∞,jis the maximum norm on interval j,h=

dk−p+1 −dk+p, and the constant Kponly depends on p, and is given by

Kp=(2p(p−1))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 f∈Ck

∆[a, b], if it has k−1 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

[dk−2, dk+3], starting from the smallest knots to the largest ones, calculate

Supk=max

j∈{2,3}K3h4||D4¯

ψj||∞,[dk−2,dk+3].(22)

If Supk> tol, where tol is the acceptable tolerance, we calculate a new knot point

d= (dk+3 +dk−2)/2. Then, we update the set of knots dby adding d, and calculate

Supkonce more time. We continue the iteration until Supk≤tol. When the condition

Supk≤tol 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 [dk−2, 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 λj∈gridλ, 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