Bene i, C is iane; Ne o, José Mon ei o Va anda; Mo i, Rogé io
A icle
Mac oeconomic de e minan s o he in e es a e e m
s uc u e: A S ensson model analysis
Economies
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Sugges ed Ci a ion: Bene i, C is iane; Ne o, José Mon ei o Va anda; Mo i, Rogé io (2025) :
Mac oeconomic de e minan s o he in e es a e e m s uc u e: A S ensson model analysis,
Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 4, pp. 1-21,
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Ci a ion: Bene i, C., Va anda Ne o, J.
M., & Mo i, R. (2025). Mac oeconomic
De e minan s o he In e es Ra e Te m
S uc u e: A S ensson Model Analysis.
Economies,13(4), 108. h ps://doi.o g/
10.3390/economies13040108
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A icle
Mac oeconomic De e minan s o he In e es Ra e Te m
S uc u e: A S ensson Model Analysis
C is iane Bene i 1,* , José Mon ei o Va anda Ne o 2and Rogé io Mo i 3
1Finance Depa men , ICN Business School, CEREFIGE, Uni e si é de Lo aine, 54000 Nancy, F ance
2Banco do No des e do B asil, Fo aleza 60743902, B azil; jose_mon ei [email p o ec ed]
3Economics Depa men , FGV EESP, Sao Paulo 01313020, B azil; oge io.mo i@ g .b
*Co espondence: [email p o ec ed]
Abs ac : This s udy de elops a model o p edic and explain sho - e m luc ua ions in
he B azilian local cu ency in e es a e e m s uc u e. The model elies on he po en ial
ela ionship be ween hese mo emen s and key mac oeconomic ac o s. The me hodology
consis s o wo s ages. Fi s , he S ensson model is applied o i he daily yield cu e da a.
This in ol es maximizing he R
2
s a is ic in an OLS eg ession, ollowing he Nelson–Siegel
app oach. The median decay pa ame e s a e hen ixed o subsequen es ima ions. In
he second s age, wi h he daily yield cu e es ima es in hand, ano he OLS eg ession is
conduc ed. This eg ession inco po a es he idea ha S ensson’s be as a e in luenced by
mac oeconomic a iables.
Keywo ds: e m s uc u e o in e es a es; S ensson model; Nelson–Siegel model; pa a-
me ic models; mac oeconomic a iables; ading algo i hms
JEL Classi ica ion: C53; C58; G17
1. In oduc ion
The in e es a e e m s uc u e, o yield cu e, is a c ucial indica o o ma ke ex-
pec a ions and cen al bank policy. By analyzing he yield cu e, in es o s can gauge
whe he in e es a es a e an icipa ed o ise, all, o emain s able. Addi ionally, i e eals
he ma ke ’s pe cep ion o he du a ion o mone a y policy ac ions. In his con ex , his
esea ch was designed wi h a ocus on de eloping and sys ema izing a model o p edic ing
sho - e m luc ua ions in he B azilian local cu ency in e es a e e m s uc u e. I aimed
o analyze he ela ionship be ween yield cu e pa ame e s using he S ensson model
(S ensson,1994) and mac oeconomic a iables, while also iden i ying and analyzing he
cons ain s a ec ing di e en ime pe iods o he in e es a e cu e (Co oneo e al.,2016).
This sys ema ic app oach made i possible o sepa a e in e es a e cu e ho izons in o
sho , medium, and long e ms while highligh ing he condi ioning ac o s o each pe iod.
The s udy posi ed ha mac oeconomic a iables ha e a signi ican in luence on yield
cu e mo emen s, wi h di e en a iables showing a ying impac s ac oss di e en ma u-
i y ho izons (as discussed by Co oneo e al.,2016). Cen al banks ely on a as a ay o
inancial and mac oeconomic da a o in o m hei decisions. Resea ch by Moench (2008),
Lud igson and Ng (2009) and Caldei a e al. (2025) highligh s he impo ance o mac oeco-
nomic a iables in p edic ing u u e yield cu e mo emen s (Baue & Rudebusch,2020).
These s udies sugges ha elying solely on cu en yield da a may lead o subop imal o e-
cas s. Beyond yield cu e analysis, ma ke pa icipan s assess in es o sen imen h ough
in e es a e ola ili y. Va ious ac o s, including he benchma k phenomenon, coupon
Economies 2025,13, 108 h ps://doi.o g/10.3390/economies13040108
Economies 2025,13, 108 2 o 21
e ec , liquidi y e ec , and ax conside a ions, con ibu e o p icing e o s in bond ma ke s.
P e ious esea ch by El on and G een (1998), Amihud and Mendelson (1991), and Chen
e al. (2024) has iden i ied hese ac o s as signi ican in luences on p icing accu acy.
This s udy aims o analyze he ela ionship be ween he B azilian in e es a e cu e
pa ame e s, as modeled by S ensson (1994), and ele an mac oeconomic a iables. By
di iding he in e es a e ho izon in o sho , medium, and long e ms, we can iden i y po-
en ial cons ain s o each pe iod. I was hypo hesized ha he S ensson model (S ensson,
1994) wi h wo decay ac o s would p o ide a supe io i compa ed o he Nelson–Siegel
model (Nelson & Siegel,1987) o B azilian yield cu es, pa icula ly in cap u ing complex
cu e dynamics. Addi ionally, he s udy heo ized ha di e en po ions o he yield cu e
(sho s. long e m) would be impac ed by dis inc mac oeconomic ac o s, e lec ing he
a ying na u e o ma ke in luences ac oss di e en ime ho izons. Sensi i i y analyses
we e conduc ed o assess he models’ alignmen wi h exis ing economic heo y.
Ou esea ch con ibu es o he exis ing li e a u e p esen ing a comp ehensi e analysis
using B3 e m s uc u e da a up o 15-yea ma u i ies in B azil, in oducing an in eg a ion
o yield cu e i ing wi h explana o y mac oeconomic a iables. A pa icula ly signi i-
can inding is ha he B azilian yield cu e peaks a app oxima ely one- hi d he ime
o he US cu e based on decay pa ame e s, ep esen ing an impo an disco e y o
eme ging ma ke yield cu e dynamics. Fu he mo e, we used a mo ing geome ic mean
index o expec ed in la ion and GDP, adding o he me hodological oolki a ailable o
such analyses.
The esea ch o e s subs an ial p ac ical alue ac oss a ious inancial sec o s and
applica ions. The model has demons a ed s ong p edic i e powe o sho e - e m ma-
u i ies, making i pa icula ly use ul o ading s a egies and ma ke analysis. I can
be e ec i ely u ilized by clea inghouses, inancial in o ma ion p o ide s, cen al banks,
and inancial ins i u ions o bo h analy ical and ope a ional pu poses. The amewo k
p o ides sys ema ic analysis o mac oeconomic impac s on yield cu es, o e ing po en ial
applica ions in algo i hmic ading wi h demons a ed success a es abo e andom chance.
Addi ionally, i se es as a aluable ool o unde s anding how di e en ac o s a ec
sho - e m e sus long- e m a es, p o iding insigh s o mone a y policy implemen a ion,
as also discussed by Ak am and Uddin (2021).
The es o he pape is s uc u ed as ollows. Sec ion 2b ie ly desc ibes he Nel-
son and Siegel (1987) and S ensson (1994) models. The es ima ion p ocedu e using he
p oposed model and he da a s uc u e is discussed in Sec ion 3. Sec ion 4p esen s he
es ima ion esul s. Finally, Sec ion 5concludes he pape and he appendix p o ides some
addi ional de ails.
2. Modeling he Te m S uc u e o In e es Ra es
2.1. Theo e ical F amewo k and Model Implemen a ion: The B azilian In e es Ra e Ma ke
S udies on he yield cu e ha e been conduc ed since he 1970s (Caldei a e al.,2015,
2025;Ullah e al.,2015;Tsai,2012), by bo h inancie s and economis s. Financie s ocus
on de i a i e p icing and ixed- a e bond hedging, while economis s seek o link he
yield cu e o mac oeconomic a iables. Non-pa ame ic models ensu e a pe ec i o
he yield cu e a a gi en poin , elimina ing a bi age oppo uni ies be ween ma u i ies.
These models a e widely used by in e es a e ade s due o hei non-a bi age p op-
e y. Examples include cubic splines (McCulloch,1971,1975) and exponen ial splines
(Vasicek,1977).
Equilib ium models, a ype o pa ame ic model, ocus on modeling he ins an aneous
a e and i s e olu ion o e ime h ough isk p emium modeling. No able examples include
Vasicek (1977), Cox e al. (1985), and Hull and Whi e (1990). As no ed by Caldei a (2011),
Economies 2025,13, 108 3 o 21
pa ame ic me hods o e se e al ad an ages. They p o ide pa simonious speci ica ions
wi h economic in e p e a ions and can impose unc ional o ms consis en wi h economic
heo y. These ea u es make hem aluable o s udying he ela ionship be ween he yield
cu e and mac oeconomic a iables (Kuma & Vi mani,2022).
In B azil di e en s udies ha e been conduc ed using he Nelson and Siegel (1987)
model (Ta anielli & Lau ini,2023), which i s he yield cu e by sol ing a di e en ial
equa ion o o wa d a es and in eg a ing hem o ob ain he spo yield cu e. I cap u es
he cu a u e cha ac e is ic o eal in e es a e cu es by in oducing a o ma pa ame e
calcula ed o maximize he R
2
measu e in an OLS eg ession. In his model, he spo cu e is
R(m)=β0+β1[1−exp(−m/τ)]/(m/τ)+β2{[1−exp(−m/τ)]/(m/τ)−exp(−m/τ)}(1)
As shown in Equa ion (1), once he pa ame e
τ
is se , he coe icien s o he inde-
penden a iables can be calcula ed o a speci ic da e. Since he second and hi d e ms
app oach ze o as ime ends o in ini y,
β0
is e e ed o as he long- e m ac o , ep esen ing
he a e o which he model con e ges. The second e m also app oaches ze o, making
β1
he sho - e m ac o . The hi d e m exhibi s an in e es ing pa e n, eaching a peak a an
in e media e poin and hen declining s eadily o ze o. This beha io iden i ies i as he
medium- e m ac o .
An al e na i e o he Equa ion (1) is by using he pa ame e =1/τ:
y (m)=β1+β21−e−λm
λm+β31−e−λm
λm−e−λm(2)
The pa ame e
λ
con ols he a e o exponen ial decay. Smalle
λ
alues esul in a
slowe decay, p o iding a be e i o longe - e m cu es. Con e sely, la ge
λ
alues lead
o as e decay, aligning be e wi h sho - e m in e es a e cu es. The alue o
λ
also
de e mines he medium- e m poin o maximum e ec , which is cap u ed by β3.
S ensson (1994) in oduced an addi ional e m o he Nelson and Siegel (1987) equa ion
o imp o e i s i in ce ain cases, such as p onounced S-cu es. This e m,
β3
, is he
independen a iable o a leg con aining he ou h e m in he equa ion, inco po a ing he
pa ame e τ2.
The equa ion in he S ensson model (S ensson,1994) is as ollows:
i(m;b)=β0+β1
1−exp−m
τ1
m
τ1
+β2
1−exp−m
τ1
m
τ1
−exp−m
τ1
+β3
1−exp−m
τ2
m
τ2
−exp−m
τ2
(3)
Diebold and Li (2006) explo ed he empo al dynamics o he yield cu e by modeling
he pa ame e s
β
as ollowing an au o eg essi e p ocess. This app oach ep esen s he
pa ame e s β, which a y o e ime, as β .
y (τ)=β1 +β2 1−e−λτ
λτ +β3 1−e−λτ
λτ −e−λτ(4)
Diebold and Li (2006) in oduced a signi ican pe spec i e by in e p e ing he h ee
ac o s as le el (
β1
), slope (
β2
), and cu a u e (
β3
). Huse (2011) ex ended he Nelson and
Siegel (1987) model by eplacing he la en ac o s o Diebold and Li (2006) wi h coe icien s
whose dynamics a e linked o obse able mac oeconomic s a e a iables, such as in la ion,
deb , and he policy a e. The equa ion go e ning he h ee- ac o , one-pa ame e model is
as ollows:
y (τi)=β1 +β2 1−e−λτi
λτi+β3 1−e−λτi
λτi
−e−λτi+u (τi)(5)
Economies 2025,13, 108 4 o 21
In Huse (2011), he ec o
θ
= (
β ′
,
λ
)
′
is de ined as he ime- a ying pa ame e
go e ning he shape o he yield cu e. This ec o is pa i ioned in o wo componen s: he
mean and he s a e a iables ha in luence he la en a iables. The Huse model (Huse,
2011) can be ep esen ed as ollows:
y (τ)=X (λ )β +u ( )(6)
wi h: "β
λ #="βmean
λmean#+M −"σβ
σλ#(7)
Acco ding o Huse (2011, p. 3243), “al hough his model is mo e expensi e o p edic
om a nume ical poin o iew han ha o Diebold and Li (2006), in he gene al case whe e
he pa ame e
λ
is also es ima ed by s a e a iables, his cos is o se by he ac ha he
dynamics o he pa ame e β is desc ibed by he s a e a iables”.
2.2. The In e es Ra e Ma ke in B azil
The B azilian in e es a e ma ke is cha ac e ized by daily in o ma ion low and
specula ion ega ding u u e in e es a e mo emen s, acili a ed by a di e se a ay o
inancial ins umen s. The p ima y ins umen s employed by ma ke pa icipan s o place
be s and shape he yield cu e a e one-day in e bank de i a i es aded on he B azilian
Me can ile & Fu u es Exchange (B3) and go e nmen bonds issued by he T easu y. Gomes
da Sil a e al. (2011) in es iga ed he in luence o mone a y au ho i y signaling on yield
cu e dynamics. In hei s udy, he ixed a e a ime o ma u i y n, exp essed as a
con inuous yield, is ob ained by
R =1
n n
∑
i=0
E +i!+ε (8)
The p eceding equa ion illus a es ha he ixed a es comp ising he yield cu e
a e equi alen o he cumula i e in e bank a e (DI) o he gi en pe iod, augmen ed by
a isk p emium.
E
ep esen s he expec ed alue ope a o a ime . When applied o
DI
,n
, i calcula es he expec ed alue o he accumula ed DI a e o e he pe iod om
o ma u i y n, based on all in o ma ion a ailable o ma ke pa icipan s a ime . This
o mula ion aligns wi h he es ablished e m s uc u e li e a u e, pa icula ly in he con ex
o how mone a y policy signals impac yield cu e mo emen s (Ak am & Uddin,2021).
The exp ession
E
[DI
,n
], combined wi h he isk p emium
π ,n
, p o ides a comp ehensi e
ep esen a ion o how ixed in e es a es a e de e mined in he B azilian ma ke .
When ading in e bank deposi u u es con ac s on he B3, ma ke pa icipan s con-
ey hei expec a ions ega ding u u e in e es a es o liquid con ac ma u i ies h ough
he in e play o supply and demand (Ta anielli & Lau ini,2023;A ie ian o e al.,2024).
This esea ch aims o de elop a model ha sys ema izes he mac oeconomic de e minan s
unde lying hese shi s in he yield cu e’s shape.
The B azilian in e es a e ma ke ope a es h ough a dual-channel sys em comp ising
one-day in e bank de i a i es aded on he B azilian Me can ile & Fu u es Exchange (B3)
and go e nmen bonds issued by he T easu y. The o ma ion o he in e es a e e m
s uc u e ollows a sys ema ic p ocess ha begins wi h he Selic a e se by he Cen al
Bank’s COPOM (Mone a y Policy Commi ee in B azil) mee ings, ex ends h ough epo
ope a ions o main ain he e ec i e a e, and con inues in o he in e bank CDI ma ke 1.
The B azilian yield cu e exhibi s se e al dis inc i e cha ac e is ics ha se i apa
om o he ma ke s, pa icula ly in i s decay ac o s and peak iming. The cu e ypically
peaks a one- hi d o he ime ho izon compa ed o he Ame ican yield cu e, wi h maxi-
Economies 2025,13, 108 5 o 21
mum s ess poin s on
β2
and
β3
occu ing a app oxima ely 8 and 7 mon hs, espec i ely.
Ma ke dynamics a e hea ily in luenced by daily in o ma ion low, COPOM decisions on
Wednesdays, and he Focus Bulle in da a published on he i s business day o each week
2
.
3. P oposed Model
3.1. Fi ing he Yield Cu e
Due o he econome ic challenges associa ed wi h he Huse model (Huse,2011), we
p opose o ecas ing using he S ensson model (S ensson,1994). This in ol es calcula ing
he pa ame e s
λ1
and
λ2
o each B3 ading session ollowing he Nelson and Siegel (1987)
p ocedu e. Subsequen ly, he median o hese esul s was used o es ima e he eg ession
be ween he pa ame e s βand ele an mac oeconomic a iables.
3.2. Da a Sample
Ou sample pe iod includes he 15-yea in e es a e e ex, which became a ailable o
ading in 2006. This longe - e m e ex is c ucial o comp ehensi e yield cu e analysis
and p o ides unique insigh s in o he B azilian ixed income ma ke ’s e m s uc u e. The
da ase includes weekly adjus ed da a o he B azilian ma ke om 2 Janua y 2006 o 8
Decembe 2014 and i comp ises 466 weeks o obse a ions, o e ing obus s a is ical powe
o ou analysis. This ex ensi e weekly da ase allows o ho ough es ing o he S ensson
model’s (S ensson,1994) pe o mance ac oss di e en ma ke condi ions, p o iding a solid
ounda ion o ou me hodological amewo k.
Reg ession Agains S a e Va iables
In addi ion o he challenges inhe en in o ecas ing econome ic models, a c ucial
conside a ion is he op imal da a equency o eg ession. Daily equency could in oduce
noise due o sho - e m luc ua ions in ma ke ac i i y and he inclusion o a iables no
cap u ed by he model. Mon hly equency, on he o he hand, migh obscu e impo an
sho - e m shi s in agen expec a ions. Consequen ly, we ha e selec ed weekly equency
o ou s a is ical and eg ession es s.
To ensu e da a consis ency, he independen a iables should ideally be upda ed
weekly. Po en ial explana o y a iables we e included (mo e in o ma ion Table 1):
Ma ke Expec a ions Va iables:
•
DIPCA-Focus: In la ion expec a ions collec ed h ough he Cen al Bank o B azil’s
Focus Ma ke Repo Su ey, based on he Ex ended Na ional Consume P ice Index
(IPCA), published by IBGE.
•
DGDP-Focus: GDP g ow h expec a ions ob ained om he Cen al Bank’s Focus
Bulle in, ep esen ing ma ke consensus o economic ac i i y.
Domes ic Economic Indica o s:
•
IPC-Fipe: The Consume P ice Index calcula ed by FIPE (Economic Resea ch Ins i-
u e Founda ion) o he São Paulo me opoli an egion, measu ing cu en in la ion,
weekly da a.
•
Selic: The B azilian basic in e es a e se by he Cen al Bank’s Mone a y Policy
Commi ee (COPOM).
•
DExchangeRa e: The nominal exchange a e be ween he B azilian Real (BRL) and US
Dolla (USD).
•Economic Ac i i y: Es ima es om he cen al bank’s Focus Bulle in.
Risk Measu es:
Economies 2025,13, 108 6 o 21
•
CDS: The 5-yea C edi De aul Swap sp ead o B azil, se ing as a measu e o
so e eign isk and coun y isk pe cep ion. Weekly da a de i ed om he 5-yea B azil
CDS, a daily quo ed indica o .
•
VIX: The CBOE Vola ili y Index, measu ing global sys emic isk h ough S&P 500
op ions implied ola ili y
In e na ional Ma ke Indica o s:
•
T-Bill 10A: The 10-yea US T easu y yield, ep esen ing he ex e nal in e es a e and
global isk- ee a e
•
CRB Index: The Commodi y Resea ch Bu eau index acking commodi y p ices,
pa icula ly ele an gi en B azil’s signi ican ag icul u al and mine al expo s
These a iables we e selec ed based on hei heo e ical and empi ical ele ance o
in e es a e de e mina ion in B azil, ollowing es ablished li e a u e in he ield. The
combina ion o domes ic and in e na ional a iables allows us o cap u e bo h local and
global ac o s a ec ing he B azilian yield cu e.
Table 1. Va iables in o ma ion.
Da a Sou ce Websi e (Accessed on
25 Feb ua y 2025) Pe iod
In e es Ra e Ma ke
B azilian Me can ile &
Fu u es Exchange (B3),
now pa o B3
www.b3.com.b /en_us/
ma ke -da a-and-indices/
Daily da a om Janua y
2006 o Decembe 2014
Ma ke Expec a ions
Expec ed In la ion (IPCA) B azilian Cen al Bank’s
Focus Ma ke Repo
www.bcb.go .b /en/
publica ions/
ocusma ke eadou
Weekly
GDP g ow h expec a ions B azilian Cen al Bank’s
Focus Ma ke Repo
www.bcb.go .b /en/
publica ions/
ocusma ke eadou
Weekly
Domes ic Economic Indica o s
Cu en In la ion IPC-FIPE (Consume P ice
Index)
www. ipe.o g.b (São
Paulo me opoli an a ea) weekly
Basic In e es Ra e (Selic) B azilian Cen al Bank
(BCB) www.bcb.go .b Daily upda es
Exchange Ra e B azilian Cen al Bank
(BCB) www.bcb.go .b Daily BRL/USD a e
Risk Measu es
Coun y Risk 5-yea B azil C edi
De aul Swap (CDS) quo es
Bloombe g Daily quo es
Global Risk
CBOE Vola ili y Index
(VIX)—Chicago Boa d
Op ions Exchange
www.cboe.com Daily
In e na ional Ma ke
Ex e nal In e es Ra e
10-yea U.S. T easu y
Bills—U.S. T easu y
Depa men
www. easu y.go Daily
Commodi y Index Co eCommodi y CRB
Index Thomson Reu e s Daily
Economies 2025,13, 108 7 o 21
All da a se ies co e he pe iod om 2 Janua y 2006, o 8 Decembe 2014, comp ising
466 weeks o obse a ions. The da a we e collec ed and p ocessed a weekly equency o
main ain consis ency ac oss all a iables (all da a we e accessed du ing 18 No embe 2015
and 18 Decembe 2015).
3.3. The Choice o λFac o s
To calcula e he daily alues o
λ1
( ) and
λ2
( ), we ollowed a simila p ocedu e as
ou lined in Nelson and Siegel (1987), employing he wo lambda pa ame e s om he
S ensson model (S ensson,1994).
Fo each a ailable da e, we ixed
λ1
and
λ2
and pe o med an OLS eg ession on he
yield cu e pa ame e s
β0
,
β1
,
β2
, and
β3
. The R
2
s a is ic was eco ded, and
λ1
( ) and
λ2
( )
we e allowed o a y wi hin an app op ia e ange. Fo each pai o
λ1
( ),
λ2
( ), a se o leas
squa es es ima o s
β0
,
β1
,
β2
, and
β3
was ob ained, along wi h he associa ed R
2
measu e.
The combina ion o
β0
,
β1
,
β2
,
β3
,
λ1
, and
λ2
yielding he highes R
2
was selec ed o model
he yield cu e a ha ime.
We op ed o a simple OLS app oach ins ead o obus op imiza ion algo i hms like
maximum likelihood due o he g ea e implemen a ion challenges and po en ial issues
wi h inding op imal solu ions. As no ed by Diebold and Li (2006), using a p ede e mined
alue o
λ
simpli ies he p ocess and enhances nume ical c edibili y by educing he need
o complex nume ical op imiza ions.
To de e mine an app op ia e alue o
λ
, we ollowed he app oach o Nelson and
Siegel (1987), selec ing he alue whe e he load o he cu a u e pa ame e eaches i s
maximum. By de i ing he load ac o
β3
as a unc ion o he pa ame e , we ob ain he
ollowing exp ession:
dc
dλ=e−λτh(λτ)2+λτ +1i−1 (9)
whe e
c=1−exp(−λτ)
λτ −exp(−λτ)(10)
The pa ame e “c” ep esen s he cu a u e ac o loading coe icien in he Nelson–
Siegel e m s uc u e amewo k. I means ha “c” is a calib a ion cons an aking he
de i a i e o he cu a u e componen wi h espec o ma u i y o ind he poin o maxi-
mum cu a u e impac . I also se es as a scaling ac o in he op imiza ion p ocess when
compa ing S ensson and Nelson–Siegel model speci ica ions (Nelson & Siegel,1987). This
cons an helps main ain consis ency in he model’s abili y o cap u e he yield cu e’s
cu a u e cha ac e is ics. The cons an “c” is pa icula ly ele an o he B azilian yield
cu e, which exhibi s unique cha ac e is ics compa ed o de eloped ma ke s. I helps
in cap u ing he cu e’s peak ha ypically occu s a app oxima ely one- hi d o he
ime ho izon.
This gi es λ= 0.0609 o τ= 30 mon hs.
In Figu e 1, he his og am o alues o he decay ac o
λ
om he Nelson and Siegel
(1987) model o he sample used in his s udy is shown below:
Economies 2025,13, 108 8 o 21
Economies 2025, 13, x FOR PEER REVIEW 8 o 22
Figu e 1. His og am 𝜆—Nelson and Siegel Model.
By subs i u ing he median alue o λ in o Equa ion (9), we calcula ed he ma u i y
a which he load o he cu a u e ac o eaches i s maximum o B azil. This alue was
ound o be app oxima ely 16 mon hs, which is oughly hal he Ame ican alue.
In Figu e 2, he his og am below illus a es he dis ibu ion o λ1 alues ob ained om
he S ensson (1994) model wi hin he sample used.
Figu e 2. His og am 𝜆—S ensson Model.
In Figu e 3, he his og am o alues o he decay ac o 𝜆 in he S ensson model
(S ensson, 1994) in he sample used is shown below:
Figu e 3. His og am 𝜆—S ensson Model.
0
200
400
600
800
1,000
0.00 0.01 0.02 0.03 0.04 0.05
Se ies: LAMBDA
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.005215
Median 0.002941
Maximum 0.050000
Minimum 0.000667
S d. De . 0.005985
Skewness 2.258207
Ku osis 8.981364
Ja que-Be a 5257.016
P obabili y 0.000000
0
100
200
300
400
500
600
700
800
0.00 0.02 0.04 0.06 0.08 0.10
Se ies: LAMBDA1
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.010108
Median 0.007143
Maximum 0.100000
Minimum 0.001000
S d. De . 0.010762
Skewness 2.983089
Ku osis 15.63886
Ja que-Be a 18280.19
P obabili y 0.000000
0
100
200
300
400
500
0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750 0.0875
Se ies: LAMBDA2
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.011155
Median 0.007092
Maximum 0.090909
Minimum 0.000999
S d. De . 0.016074
Skewness 3.579309
Ku osis 17.03443
Ja que-Be a 23228.42
P obabili y 0.000000
Figu e 1. His og am λ—Nelson and Siegel Model.
By subs i u ing he median alue o
λ
in o Equa ion (9), we calcula ed he ma u i y
a which he load o he cu a u e ac o eaches i s maximum o B azil. This alue was
ound o be app oxima ely 16 mon hs, which is oughly hal he Ame ican alue.
In Figu e 2, he his og am below illus a es he dis ibu ion o
λ1
alues ob ained om
he S ensson (1994) model wi hin he sample used.
Economies 2025, 13, x FOR PEER REVIEW 8 o 22
Figu e 1. His og am 𝜆—Nelson and Siegel Model.
By subs i u ing he median alue o λ in o Equa ion (9), we calcula ed he ma u i y
a which he load o he cu a u e ac o eaches i s maximum o B azil. This alue was
ound o be app oxima ely 16 mon hs, which is oughly hal he Ame ican alue.
In Figu e 2, he his og am below illus a es he dis ibu ion o λ1 alues ob ained om
he S ensson (1994) model wi hin he sample used.
Figu e 2. His og am 𝜆—S ensson Model.
In Figu e 3, he his og am o alues o he decay ac o 𝜆 in he S ensson model
(S ensson, 1994) in he sample used is shown below:
Figu e 3. His og am 𝜆—S ensson Model.
0
200
400
600
800
1,000
0.00 0.01 0.02 0.03 0.04 0.05
Se ies: LAMBDA
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.005215
Median 0.002941
Maximum 0.050000
Minimum 0.000667
S d. De . 0.005985
Skewness 2.258207
Ku osis 8.981364
Ja que-Be a 5257.016
P obabili y 0.000000
0
100
200
300
400
500
600
700
800
0.00 0.02 0.04 0.06 0.08 0.10
Se ies: LAMBDA1
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.010108
Median 0.007143
Maximum 0.100000
Minimum 0.001000
S d. De . 0.010762
Skewness 2.983089
Ku osis 15.63886
Ja que-Be a 18280.19
P obabili y 0.000000
0
100
200
300
400
500
0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750 0.0875
Se ies: LAMBDA2
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.011155
Median 0.007092
Maximum 0.090909
Minimum 0.000999
S d. De . 0.016074
Skewness 3.579309
Ku osis 17.03443
Ja que-Be a 23228.42
P obabili y 0.000000
Figu e 2. His og am λ1—S ensson Model.
In Figu e 3, he his og am o alues o he decay ac o
λ2
in he S ensson model
(S ensson,1994) in he sample used is shown below:
Economies 2025, 13, x FOR PEER REVIEW 8 o 22
Figu e 1. His og am 𝜆—Nelson and Siegel Model.
By subs i u ing he median alue o λ in o Equa ion (9), we calcula ed he ma u i y
a which he load o he cu a u e ac o eaches i s maximum o B azil. This alue was
ound o be app oxima ely 16 mon hs, which is oughly hal he Ame ican alue.
In Figu e 2, he his og am below illus a es he dis ibu ion o λ1 alues ob ained om
he S ensson (1994) model wi hin he sample used.
Figu e 2. His og am 𝜆—S ensson Model.
In Figu e 3, he his og am o alues o he decay ac o 𝜆 in he S ensson model
(S ensson, 1994) in he sample used is shown below:
Figu e 3. His og am 𝜆—S ensson Model.
0
200
400
600
800
1,000
0.00 0.01 0.02 0.03 0.04 0.05
Se ies: LAMBDA
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.005215
Median 0.002941
Maximum 0.050000
Minimum 0.000667
S d. De . 0.005985
Skewness 2.258207
Ku osis 8.981364
Ja que-Be a 5257.016
P obabili y 0.000000
0
100
200
300
400
500
600
700
800
0.00 0.02 0.04 0.06 0.08 0.10
Se ies: LAMBDA1
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.010108
Median 0.007143
Maximum 0.100000
Minimum 0.001000
S d. De . 0.010762
Skewness 2.983089
Ku osis 15.63886
Ja que-Be a 18280.19
P obabili y 0.000000
0
100
200
300
400
500
0.0000 0.0125 0.0250 0.0375 0.0500 0.0625 0.0750 0.0875
Se ies: LAMBDA2
Sample 1/02/2006 12/08/2014
Obse a ions 2246
Mean 0.011155
Median 0.007092
Maximum 0.090909
Minimum 0.000999
S d. De . 0.016074
Skewness 3.579309
Ku osis 17.03443
Ja que-Be a 23228.42
P obabili y 0.000000
Figu e 3. His og am λ2—S ensson Model.
Economies 2025,13, 108 15 o 21
Economies 2025, 13, x FOR PEER REVIEW 15 o 22
Figu e 7. p- alues o hypo hesis es s on he e ices.
The analysis shown in Figu e 7 e ealed he s onges p edic i e pe o mances a 252
days (p- alue = 1.22%) and 294 days (p- alue = 2.59%). Addi ionally, i showed a s a is i-
cally signi ican p edic i e powe o e ices be ween 168 and 399 days, and inally, he
p- alues below 0.10 indica e pe o mance signi ican ly be e han andom o sho /me-
dium- e m ma u i ies.
Table 6 con ains a summa y o he e ices a which he model bea s he ossing o a
coin o a p- alue wi h a limi o 10%.
Table 6. Rele an p- alues o he model a he e ices.
Ve ex
168 210 252 273 294 315 336 357 378 399
p-Value 6.33% 9.47% 1.22% 5.12% 2.59% 5.12% 3.27% 5.12% 6.33% 6.33%
The model’s ou pe o mance a e ices be ween 168 and 399 days is pa icula ly sig-
ni ican because hese ma u i ies ep esen he mos ac i ely aded segmen s o he B a-
zilian in e es a e u u es ma ke . The s onges p edic i e powe occu s a 252 and 294
days (wi h p- alues o 1.22% and 2.59%, espec i ely), which p ecisely aligns wi h ypical
in es men ho izons in he B azilian ixed income ma ke . Mo eo e , hese e ices co e-
spond o he mos liquid segmen s o he B azilian in e es a e u u es ma ke , whe e
p ice disco e y is mos efficien . The model’s effec i eness in hese segmen s is pa icu-
la ly aluable as hey ep esen ma u i ies whe e mos ma ke ac i i y occu s, making he
p edic ions p ac ically applicable o ma ke pa icipan s.
The p edic i e powe aligns wi h B azil’s mone a y policy ansmission ho izon,
making he indings aluable o policy analysis. The esul s a e pa icula ly ele an o
analyzing ma ke eac ions a ound COPOM (Mone a y Policy Commi ee) mee ings, as
hese e ices cap u e bo h immedia e and medium- e m policy impac s. These esul s
also co obo a e Ak am and Uddin (2021).
The indings ha e p ac ical applica ions o he de elopmen o ading algo i hms
ocused on medium- e m a e mo emen s; he use o isk managemen ools o inancial
ins i u ions; ma ke analysis a ound mone a y policy e en s; and suppo o clea ing-
houses and inancial in o ma ion p o ide s equi ing eliable a e p edic ions.
Ano he highligh is ha he a ying pe o mance ac oss e ices e lec s ma ke mi-
c os uc u e aspec s, such as e y sho e ices (unde 31.5 business days), and shows
less su p ise due o mone a y policy mee ing in e als. On he o he hand, longe - e m
Figu e 7. p- alues o hypo hesis es s on he e ices.
The analysis shown in Figu e 7 e ealed he s onges p edic i e pe o mances a
252 days (p- alue = 1.22%) and 294 days (p- alue = 2.59%). Addi ionally, i showed
a s a is ically signi ican p edic i e powe o e ices be ween 168 and 399 days, and
inally, he p- alues below 0.10 indica e pe o mance signi ican ly be e han andom o
sho /medium- e m ma u i ies.
Table 6con ains a summa y o he e ices a which he model bea s he ossing o a
coin o a p- alue wi h a limi o 10%.
Table 6. Rele an p- alues o he model a he e ices.
Ve ex
168 210 252 273 294 315 336 357 378 399
p-Value 6.33% 9.47% 1.22% 5.12% 2.59% 5.12% 3.27% 5.12% 6.33% 6.33%
The model’s ou pe o mance a e ices be ween 168 and 399 days is pa icula ly
signi ican because hese ma u i ies ep esen he mos ac i ely aded segmen s o he
B azilian in e es a e u u es ma ke . The s onges p edic i e powe occu s a 252 and
294 days (wi h p- alues o 1.22% and 2.59%, espec i ely), which p ecisely aligns wi h
ypical in es men ho izons in he B azilian ixed income ma ke . Mo eo e , hese e ices
co espond o he mos liquid segmen s o he B azilian in e es a e u u es ma ke , whe e
p ice disco e y is mos e icien . The model’s e ec i eness in hese segmen s is pa icula ly
aluable as hey ep esen ma u i ies whe e mos ma ke ac i i y occu s, making he
p edic ions p ac ically applicable o ma ke pa icipan s.
The p edic i e powe aligns wi h B azil’s mone a y policy ansmission ho izon,
making he indings aluable o policy analysis. The esul s a e pa icula ly ele an o
analyzing ma ke eac ions a ound COPOM (Mone a y Policy Commi ee) mee ings, as
hese e ices cap u e bo h immedia e and medium- e m policy impac s. These esul s also
co obo a e Ak am and Uddin (2021).
The indings ha e p ac ical applica ions o he de elopmen o ading algo i hms
ocused on medium- e m a e mo emen s; he use o isk managemen ools o inancial in-
s i u ions; ma ke analysis a ound mone a y policy e en s; and suppo o clea inghouses
and inancial in o ma ion p o ide s equi ing eliable a e p edic ions.
Economies 2025,13, 108 16 o 21
Ano he highligh is ha he a ying pe o mance ac oss e ices e lec s ma ke
mic os uc u e aspec s, such as e y sho e ices (unde 31.5 business days), and shows
less su p ise due o mone a y policy mee ing in e als. On he o he hand, longe - e m
e ices su e om lowe liquidi y and ely on in e pola ion. The mos eliable esul s
appea in e ices wi h highe ma ke liquidi y.
4.4. Sensi i i y Analysis
I is easonable o assume ha he impac o mac oeconomic a iables on he yield
cu e a ies ac oss di e en ma u i y ho izons. The p oposed model can be used o
quali a i ely assess his phenomenon.
To examine how he mac oeconomic en i onmen in luences he yield cu e o e
ime, we ocused on he yea 2014, he mos ecen yea wi hin ou sample. Table 7
p esen s he eg ession coe icien s o he S ensson model pa ame e s (
β0
,
β1
,
β2
, and
β3
),
speci ically o he yea 2014, which se es as he baseline o sensi i i y analysis. The
able demons a es how di e en mac oeconomic a iables in luence each pa ame e o he
yield cu e.
Table 7. Model pa ame e s o 2014.
Mac o Va iable β0β1β2β3
CDS 0.00007 −0.00009 −0.00182 0.00172
Selic 0.00000 1.02517 4.64674 −4.45146
Dselic 0.38868 −0.61309 6.41705 −6.00718
DExchangeRa e 0.02445 −0.02627 0.00000 0.00000
DIPCA_Focus 4.54273 −3.91249 48.66481 −52.56513
DGDP_Focus 1.84810 −1.65663 0.00000 0.00000
D2014 0.10578 −0.10569 −0.18254 0.16401
The Long- e m Fac o (
β0
) showed sensi i i y o long- e m economic indica o s and
was a p ima y d i e o he o e all in e es a e le el. The Sho - e m Fac o (
β1
) de-
e mined he slope o he yield cu e and e lec ed immedia e mone a y policy impac s.
The Fi s Cu a u e Fac o (
β2
) explained abou 69% o cu a u e a ia ions (R
2
= 0.69),
signi ican ly in luenced by he Selic a e le el (posi i e ela ionship), CDS le el (posi i e
ela ionship), and expec ed in la ion changes (nega i e ela ionship). And inally, he sec-
ond cu a u e Fac o (
β3
) showed simila beha io o
β2
, also explained app oxima ely
71% o a ia ions (R
2
= 0.71), was a ec ed by he same mac oeconomic a iables as
β2
, and
p o ided addi ional lexibili y in cu e shape modelling.
Table 8p esen s he “Va iable Vec o ” con aining mac oeconomic da a o he speci ic
week o 26 Sep embe h ough 2 Oc obe 2014, which was used o es ima e he Yield Cu e
o 3 Oc obe 2014.
Table 8. Va iable ec o o sensi i i y es .
Mac o Va iable CDS Selic Dselic
DExchangeRa e
DIPCA_Focus
DGDP_Focus
D2014
Value 173.05 10.35% 0 0.0608 0.010% 0.010% 1
The key componen s p esen ed a Table 8a e he mac oeconomic a iables. I can
be obse ed ha he Selic a e was 10.90% pe annum (10.35% in con inuous e ms),
he exchange a e was an a e age o BRL 2.455/USD, he weekly a ia ion was BRL
0.0608/USD, he CDS sp ead was 173.05 basis poin s, he in la ion expec a ion was 6.31%
(Focus Bulle in), he weekly change inc eased in 0.01 pe cen age poin s, and he GDP
expec a ion was 0.92% (Focus Bulle in). The sensi i i y es ing showed how he yield cu e
Economies 2025,13, 108 17 o 21
esponds o mac oeconomic changes; mo eo e , i demons a es he model’s p edic i e
capabili ies, and i alida ed heo e ical ela ionships be ween a iables.
When analyzing bo h ables oge he (Tables 7and 8) i can be obse ed ha he model
demons a ed how mac oeconomic a iables in luence yield cu e pa ame e s, showed he
model’s abili y o cap u e ma ke dynamics, and p o ided empi ical suppo o heo e ical
ela ionships. This model o e s a amewo k o yield cu e es ima ion, shows how o
inco po a e mac oeconomic da a in o p edic ions, and p o ides a ool o ma ke analysis
and o ecas ing. Fu he mo e, he isk assessmen aspec helps unde s and sensi i i y o
a ious economic ac o s, allows o scena io analysis and s ess es ing, and suppo s isk
managemen decisions.
Table 9p esen s he calcula ed be a pa ame e s (
β0
,
β1
,
β2
, and
β3
) o 3 Oc obe 2014,
which we e de i ed by mul iplying he mac oeconomic a iables om Table 8wi h hei
co esponding eg ession coe icien s om Table 7. This able demons a es he p ac ical
applica ion o he S ensson model in es ima ing he yield cu e.
Table 9. Be as pa ame e s o 3 Oc obe 2014.
Be a Pa ame e s ze
β012.03%
β1−1.66%
β2−1.19%
β3−0.50%
The pa ame e s p esen ed in Table 9show ha he Long- e m Fac o (
β0
= 0.1198)
ep esen s he asymp o ic le el o he yield cu e and indica es he long- e m in e es
a e le el, and he alue e lec s he combined impac o in la ion expec a ions, GDP
g ow h expec a ions, and so e eign isk p emiums. The Sho - e m Fac o (
β1
=
−
0.0087)
de e mines he slope o he yield cu e and he nega i e alue indica es he downwa d
p essu e on sho - e m a es and he po en ial la ening o he yield cu e and is p ima ily
in luenced by mone a y policy s ance. Subsequen ly, he i s cu a u e ac o (
β2
=
−
0.0281)
con ols medium- e m cu e dynamics and he nega i e alue sugges s a downwa d
p essu e on medium- e m a es, an adjus men in he cu e’s hump shape and e lec s
ma ke expec a ions and isk p emiums. The second cu a u e ac o (
β3
= 0.0187) p o ides
addi ional lexibili y in cu e shape, he posi i e alue indica es an upwa d adjus men
in longe - e m segmen s, and he ine uning o he cu e’s shape helps cap u e complex
yield cu e pa e ns.
In ela ion o he model, i can be obse ed ha hese pa ame e s a e used o gene a e
he es ima ed yield cu e o 3 Oc obe 2014. The esul ing cu e is compa ed wi h ac ual
ma ke obse a ions, and i demons a es he model’s p ac ical e ec i eness in yield cu e
es ima ion. Mo eo e , he model’s pe o mance achie ed high goodness o i , wi h R
2
alues abo e 0.95 and acking e o s emaining wi hin app oxima ely 0.7% annually o
longe ma u i ies, and i cap u es bo h he shape and le el o he yield cu e.
Finally, he model sugges s ha he ins an aneous a e (
β0
+
β1
) showed limi ed
dependence on ma ke liquidi y condi ions. The sho - e m a es a e p ima ily d i en by
changes in expec ed in la ion, GDP g ow h expec a ions, and mone a y policy decisions.
The be a pa ame e s in Table 9gene a ed he es ima ed yield cu e shown in Figu e 8.
This compa ison wi h obse ed a es alida es he model’s accu acy and demons a es he
p ac ical u ili y o he S ensson app oach. This comp ehensi e analysis in Table 9shows
how he model e ec i ely ansla es mac oeconomic condi ions in o yield cu e pa ame e s,
p o iding bo h heo e ical insigh s and p ac ical applica ions o ma ke pa icipan s.
Economies 2025,13, 108 18 o 21
We gene a ed he es ima ed Yield Cu e below, plo ed oge he wi h he Yield Cu e
obse ed on 3 Oc obe 2014:
Economies 2025, 13, x FOR PEER REVIEW 18 o 22
Figu e 8. Es ima ed and obse ed yield cu e on 3 Oc obe 2014.
The compa ison be ween es ima ed and obse ed yield cu es e eals ha he model
achie es high goodness o i , wi h R2 alues a e aging abo e 0.95, acking e o s emain-
ing wi hin app oxima ely 0.7% annually o longe ma u i ies (68% con idence in e al),
and he model success ully cap u ing bo h he gene al shape and le el o he yield cu e.
The sensi i i y analysis demons a es how he yield cu e esponds o a ious mac-
oeconomic shocks, such as (a) a 100 basis poin inc ease in Selic a e p ima ily affec s he
cu e’s le el; (b) a 10% cu ency dep ecia ion impac s bo h slope and cu a u e; (c) a 3%
inc ease in CDS sp eads in luences he isk p emium componen , (d) a 50 basis poin in-
c ease in in la ion expec a ions affec s he o wa d a e s uc u e; and (e) a 50 basis poin
inc ease in expec ed GDP impac s he medium- e m segmen .
This analysis e eals ha he in e es a e le el is in luenced by Selic a e, exchange
a e a ia ions, CDS le el, and a ia ions in expec ed GDP and in la ion. Addi ionally, he
slope co ela es posi i ely wi h a ia ions in in la ion, GDP, CDS, and expec ed a es; and
he ins an aneous in e es a e is p ima ily in luenced by expec ed in la ion and GDP a -
ia ions. Finally, he cu a u e inc eases wi h Selic a e and dec eases wi h posi i e a ia-
ions in expec ed in la ion.
The obus ness o ou indings is suppo ed by a comp ehensi e body o e idence
ga he ed h ough igo ous es ing and analysis. We u ilized a 15-yea ime se ies o yield
cu e da a om B3, wi h he model showing supe io pe o mance (measu ed by R2) com-
pa ed o he o iginal Nelson–Siegel model. S a is ical es s demons a ed be e han an-
dom p edic i e abili y o ce ain e ices, pa icula ly in egions o g ea e s a is ical sig-
ni icance. Ou esul s aligned wi h con en ional economic heo y, despi e some esidual
au oco ela ion, and p o ided empi ical alida ion o how a iables like SELIC a e, ex-
change a e a ia ions, CDS le els, and expec ed GDP/in la ion impac diffe en aspec s
o he in e es a e cu e. Mo eo e , i includes back- es ing wi hin he sample, analysis
o p edic ion accu acy compa ed o andom ou comes, and sensi i i y analysis o diffe -
en mac oeconomic scena ios. These elemen s collec i ely demons a e ha while he da a
pe iod may no be cu en , he me hodological con ibu ions and s uc u al ela ionships
10.0%
10.5%
11.0%
11.5%
12.0%
12.5%
Ma u i y in Business Days
Ac ual x Es ima ed Compa ison
Ac ual In e es Ra e Es ima ed In e es Ra e
Figu e 8. Es ima ed and obse ed yield cu e on 3 Oc obe 2014.
The compa ison be ween es ima ed and obse ed yield cu es e eals ha he model
achie es high goodness o i , wi h R
2
alues a e aging abo e 0.95, acking e o s emain-
ing wi hin app oxima ely 0.7% annually o longe ma u i ies (68% con idence in e al),
and he model success ully cap u ing bo h he gene al shape and le el o he yield cu e.
The sensi i i y analysis demons a es how he yield cu e esponds o a ious mac oe-
conomic shocks, such as (a) a 100 basis poin inc ease in Selic a e p ima ily a ec s he
cu e’s le el; (b) a 10% cu ency dep ecia ion impac s bo h slope and cu a u e; (c) a
3% inc ease in CDS sp eads in luences he isk p emium componen , (d) a 50 basis poin
inc ease in in la ion expec a ions a ec s he o wa d a e s uc u e; and (e) a 50 basis poin
inc ease in expec ed GDP impac s he medium- e m segmen .
This analysis e eals ha he in e es a e le el is in luenced by Selic a e, exchange
a e a ia ions, CDS le el, and a ia ions in expec ed GDP and in la ion. Addi ionally,
he slope co ela es posi i ely wi h a ia ions in in la ion, GDP, CDS, and expec ed a es;
and he ins an aneous in e es a e is p ima ily in luenced by expec ed in la ion and GDP
a ia ions. Finally, he cu a u e inc eases wi h Selic a e and dec eases wi h posi i e
a ia ions in expec ed in la ion.
The obus ness o ou indings is suppo ed by a comp ehensi e body o e idence
ga he ed h ough igo ous es ing and analysis. We u ilized a 15-yea ime se ies o yield
cu e da a om B3, wi h he model showing supe io pe o mance (measu ed by R
2
)
compa ed o he o iginal Nelson–Siegel model. S a is ical es s demons a ed be e han
andom p edic i e abili y o ce ain e ices, pa icula ly in egions o g ea e s a is ical
signi icance. Ou esul s aligned wi h con en ional economic heo y, despi e some esidual
au oco ela ion, and p o ided empi ical alida ion o how a iables like SELIC a e, ex-
change a e a ia ions, CDS le els, and expec ed GDP/in la ion impac di e en aspec s o
he in e es a e cu e. Mo eo e , i includes back- es ing wi hin he sample, analysis o
p edic ion accu acy compa ed o andom ou comes, and sensi i i y analysis o di e en
mac oeconomic scena ios. These elemen s collec i ely demons a e ha while he da a
Economies 2025,13, 108 19 o 21
pe iod may no be cu en , he me hodological con ibu ions and s uc u al ela ionships
iden i ied in ou s udy emain aluable o unde s anding and analyzing B azilian in e es
a e e m s uc u e.
5. Conclusions
Pa simonious models o i ing yield cu es a e well-es ablished ools in ma ke
inance, employed by clea ing houses, inancial in o ma ion p o ide s, cen al banks,
and inancial ins i u ions. The app oach p esen ed in his s udy, which in ol es in-
eg a ing yield cu e i ing wi h explana o y a iables, emains a subjec o ongoing
academic in e es .
Rega ding he esul s ob ained, se e al obse a ions can be made, as while eg ession
analyses gene ally p oduce sa is ac o y explana o y models, au oco ela ion o esiduals
was e iden in all cases. Howe e , he s ylized ac s and signals om he es ima o s closely
aligned wi h con en ional economic heo y. Mo eo e , he eg essions ela ing cu e
pa ame e s o mac oeconomic a iables equi ed he inclusion o yea -speci ic dummy a i-
ables. This sugges s he p esence o liquidi y condi ions and o he ac o s no cap u ed by
quan i a i e o quali a i e a iables. Fu u e s udies could explo e he impac o de i a i e
ading olume on he B3 ma ke .
Acco ding o he model, he le el o in e es a es (long- e m in e es a es) is in lu-
enced by he Selic a e, exchange a e a ia ions, CDS le el, and a ia ions in expec ed
GDP and in la ion a es. An inc ease in any o hese ac o s leads o an inc ease in he yield
cu e le el. The slope o he yield cu e is posi i ely co ela ed wi h a ia ions in in la ion,
GDP, CDS, and expec ed GDP and in la ion a es. A posi i e in la ion a ia ion ends o
yield a posi i ely sloped cu e, while a highe Selic a e has a nega i e impac on he slope.
The ins an aneous in e es a e, a combina ion o le el and slope pa ame e s, is p ima -
ily in luenced by a ia ions in expec ed in la ion and GDP, bu wi h less in ensi y han
long- e m in e es a es. In e es ingly, i is no a ec ed by liquidi y condi ions. Long- e m
a es a y ac oss di e en yea s and he join cu a u e, de e mined by he combina ion o
λ1
and
λ2
, inc eases wi h he Selic a e and dec eases wi h posi i e a ia ions in expec ed
in la ion. I is no a ec ed by a ia ions in expec ed GDP.
This esea ch aimed o model he agg ega e beha io o agen s in ol ed in in e es a e
u u es con ac s and o he ins umen s used o p ice he B azilian yield cu e, u ilizing
publicly a ailable mac oeconomic a iables ele an o he Cen al Bank o B azil’s decision-
making p ocess ega ding in e es a es. The S ensson (1994) model, a e inemen o Nelson–
Siegel’s o iginal echnique (Nelson & Siegel,1987), was employed o i he yield cu e o
eal da a. The eg ession equa ions ob ained yielded p omising esul s as he ela ionships
be ween he model’s pa ame e s and mac oeconomic a iables align wi h expec a ions
om economic and inancial heo y. Finally, he model demons a ed p edic i e powe
o ce ain e ices, ou pe o ming pu ely andom e en s. This p edic i e powe was
pa icula ly s ong o sho e - e m ma u i ies, sugges ing ha he analysis could be u he
enhanced by ocusing on a sho e ime ho izon and inco po a ing addi ional echnical
a iables.
The p ima y con ibu ion o ou pape is me hodological, demons a ing how he
S ensson model can be e ec i ely applied o eme ging ma ke s like B azil. The s uc u al
ela ionships we iden i y be ween mac oeconomic a iables and e m s uc u e emain
ele an ega dless o he speci ic ime. The undamen al economic ela ionships and
ma ke dynamics we analyze con inue o hold ue, as e idenced by he model’s abili y
o cap u e how he yield cu e esponds o changes in key a iables such as Selic a e,
exchange a e luc ua ions, CDS le els, and expec ed GDP/in la ion a es.
Economies 2025,13, 108 20 o 21
Fu he mo e, he con ibu ions o his esea ch emphasize he ma ke and he eco-
nomic academy. F om a ma ke pe spec i e his s udy is he i s o u ilize B3 e m s uc u e
da a om up o 15-yea ma u i ies in B azil, documen ing lowe p ecision in longe ma-
u i ies due o low liquidi y. I also demons a ed he S ensson model’s supe io abili y
o cap u e yield cu e sinuosi y compa ed o he Nelson–Siegel model, pa icula ly due
o he second pa ame e . Mo eo e , he pape sugges ed ha he B azilian yield cu e
peaks a app oxima ely one- hi d he ime o he US cu e based on decay pa ame e s.
The model showed good p edic i e powe , speci ically in key ading ma u i ies (be ween
i s mee ing and one yea ). F om an economic pe spec i e he esul s in his esea ch
documen ed ela ionships be ween mac oeconomic a iables and yield cu e pa ame e s
ha align wi h economic heo y. Mo eo e , he esea ch p o ided empi ical e idence
on how he B azilian yield cu e esponds o changes in policy a e, exchange a e, and
in la ion expec a ions.
Fo u u e esea ch, he ollowing modi ica ions could po en ially imp o e model
pe o mance: selec ing he op imal
λ1
and
λ2
alues and conside ing plo ing hei join
dis ibu ion o R
2
alues. Fu he mo e, a es ic ion o he model o pe iods wi h highe
liquidi y, as liquidi y cons ain s in ce ain e ices, especially longe ones, can a ec he
model’s p edic i e powe . Also, we can inco po a e addi ional a iables such as he olume
o posi ions in B3 de i a i es ma ke s, ocusing on one-day in e bank deposi s. Analyzing
he gap be ween o eign and local in es o s migh also be insigh ul.
Au ho Con ibu ions: Concep ualiza ion, J.M.V.N. and C.B.; me hodology, J.M.V.N.; so wa e,
J.M.V.N.; alida ion, J.M.V.N., C.B. and R.M.; o mal analysis, J.M.V.N. and C.B.; in es iga ion,
J.M.V.N. and C.B.; esou ces, J.M.V.N.; w i ing—o iginal d a p epa a ion, J.M.V.N. and C.B.; w i ing—
e iew and edi ing, J.M.V.N., C.B. and R.M.; isualiza ion, J.M.V.N. and C.B.; supe ision, R.M.;
p ojec adminis a ion, J.M.V.N. and C.B. All au ho s ha e ead and ag eed o he published e sion
o he manusc ip .
Funding: This esea ch ecei ed no ex e nal unding.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : The o iginal con ibu ions p esen ed in his s udy a e included in he
a icle. Fu he inqui ies can be di ec ed o he co esponding au ho . Table 1p esen s he websi es
whe e he public da a can be ound.
Con lic s o In e es : No po en ial con lic o in e es was epo ed by he au ho s.
No es
1
This public da abase can be accessed a B azilian Cen al Bank’s ime se ies da abase (www.bcb.go .b ), he B azilian T easu y’s
yield cu e da a (www. esou odi e o.com.b ), and B3’s ma ke da a po al (www.b3.com.b ). Accessed du ing 18 No embe
2015 o 18 Decembe 2015.
2
This public da abase can be accessed a Cen al Bank’s Focus Ma ke Repo (www.bcb.go .b /en/ ocus), B3 BOVESPA’s
de i a i es ading da a, and Bank o In e na ional Se lemen s (www.bis.o g) o global ma ke s uc u e analysis. Accessed on
25 Feb ua y 2025.
3
Mo e in o ma ion abou COPOM (Mone a y Policy Commi ee in B azil) can be see a h ps://www.bcb.go .b /en/
mone a ypolicy/commi ee.
4
The me hodology employs explana o y a iables obse ed a ime –1 o p edic alues a ime . This app oach helps mi iga e
po en ial endogenei y issues in he model. The use o lagged a iables necessa ily educes he e ec i e sample size by one
obse a ion. Mo eo e , his me hodology equi es using Thu sday’s ma ke close da a o p edic F iday’s alues. This empo al
alignmen s uc u e esul s in he loss o one addi ional obse a ion. This app oach ensu es ha all ele an in o ma ion is
p ope ly inco po a ed in o he model.
Economies 2025,13, 108 21 o 21
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people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
