Affine cascade models for term structure dynamics of sovereign yield curves

Sigma Variant Cascade Model for Term
Structure of Interest Rates in Colombia
Laura Natalia Lopez (Precia)
Rafael Serrano (URosario)
Sep 20th - 2019
1

Contents
Introduction
Motivation
Literature Review
A Cascade Model for the Interest Rate
Multifrequency Cascade
Factors Representation
Bond Prices and Risk Premia
Bond Prices and Constant Risk Premia
Dimension-Invariant Term Structures
Base speciﬁcation
Data and Estimation
Empirical Results
Conclusions
2

Introduction
Motivation
Literature Review
Data and Estimation
Empirical Results
Conclusions
3

Term structure of interest rates and the yield curve
Yield (to maturity) of a ﬁxed-income instrument or debt contract is the
internal rate of return (IRR) on cash ﬂows of the contract.
The term structure (TS) of interest rates is the mathematical relation-
ship between yields across different terms or maturities (2 month, 2
year, 20 year, etc. ...) for similar debt contracts.
The yield curve (YC) is the plot or graphical representation of the TS
of certain debt contracts in a given currency.
4

Colombian sovereign yields
2 4 6 8 10 12 14
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Colombian Term Structure Yield
2018-12-28
2017-04-14
2015-05-15
2013-06-14
2011-07-15
2009-08-14
2007-09-14
2005-10-14
5

Why is the YC important?
• Pricing and valuation of other financial instruments and cash
flows
• Benchmark for determining lending and savings rates in the
credit market
• Relationship to business cycle and economic growth.
Indeed, the slope of the yield curve (e.g. difference between 10-
year Treasury bond rate and the 3-month Treasury) is one of the
most powerful predictors of future economic growth, inflation, and
recessions.
A positively sloped yield curve is often an indicator of inflationary
growth. An inverted yield curve is often an indicator of recession.
6

What model?
In order to model the TS, we need to choose ﬁrst the quantity that
serves as state variable and can help determine the dynamics of TS
from observed yields.
8

What model?
We choose the instantaneous short (-term) rate.
8

What model?
For its deﬁnition, we assume there exists a continuum of zero-coupon
(ZC) bonds, one for each positive term-to-maturity (TTM) τ > 0
• Time measured in years
• P(t, τ) = price at time t of a ZC bond with TTM τ > 0
• y(t, τ) = yield-to-maturity (YTM) deﬁned as the (continuously
compounded, annualized) IRR of the ZC bond
P(t, τ) = e−y(t,τ)τ
8

What model?
For its deﬁnition, we assume there exists a continuum of zero-coupon
(ZC) bonds, one for each positive term-to-maturity (TTM) τ > 0
• Time measured in years
• P(t, τ) = price at time t of a ZC bond with TTM τ > 0
• y(t, τ) = yield-to-maturity (YTM) deﬁned as the (continuously
compounded, annualized) IRR of the ZC bond
P(t, τ) = e−y(t,τ)τ
• Short rate rt := lim∆t→0 y(t, ∆t)
8

Why short rate?
• 1st generation of stochastic interest rate models use the
instantaneous short rate as the state variable.
• Two key advantages: very simple yet highly tractable, as they
often lead to analytic formulae for bonds and other associated
ﬁnancial instruments.
• Drawback: instantaneous short rate is a mathematical idealisation
rather than something that can be observed directly in the market.
In the past decade, short-rate models have to some extent, been
superseded by the LIBOR or SWAP market models in which the
stochastic state variable is a set of benchmark forward LIBOR or
SWAP rates.
Nonetheless, short-rate models remain still very popular among
both academic and practitioners.
9

Contribution
• We compare two short-rate models of recursive cascade with
heterogeneously persistent factors that revert in mean to the
immediate previous factor.
• Under certain speciﬁcations, the number of parameters is
invariant to the number of cascade factors.
• These models are adjusted to determine the dynamics of the
term structure of interest rates in Colombia.
• The standard model has an assumption in the volatility of the
model that always makes it constant.
• The sigma variant model (SV), proposed in this work, relaxes this
assumption by giving it a structure dependent on the size of the
cascade.
10

Literature
• Arbitrage free TSM’s (Duffie and Kan (1996), affine
specifications; Dai and Singlenton (2000, 2002))
• Low-dimensional DTSM’s, limitations of a model with only one
factor (Duan and Simonato (1999), Bolder (2001), Zeytun and
Gupta (2007))
• DTSM’s have problems in dimensionality and parameter
identification (Duffee and Stanton (2008); Duffe (2011))
11

Literature
• The 2-factor models have different conclusions (Nowman (2010)
vs. Walther (2003)). While Chatterjee (2005) concludes that a
three-factor model is adequate.
• Calvet, Laurent and Fisher (2018) propose a multifactorial
cascade model that under certain assumptions maintains its
dimension invariant.
• In Colombia there have been no studies beyond 3 factors.
Rodriguez (2016) concludes that 1-factor Vasicek model has a
poor forecast ﬁt and Velsquez et al (2016) 3-factor model ﬁts well
the curve.
12

Introduction
Factors Representation
Bond Prices and Risk Premia
Bond Prices and Constant Risk Premia
Data and Estimation
Empirical Results
Conclusions
13

Afﬁne TS model
A short rate model is said to provide an afﬁne TS if the prices of the
ZC bonds have the form
P(t, τ) = exp[−b(τ)T
Xt − c(τ)]
with Xt a multivariate stochastic process that models macro- and mi-
croeconomic unobservable variables that affect the yield curve across
terms (short, medium and long)
14

Cascade afﬁne TS model
Model assumptions
• Time is continuous with an inﬁnite horizon t ∈ [0, ∞).
• Probability Space {Ω, F, P, (Ft )t 0}.
• P is the measure of physical probability.
• Wt = (W1,t , ..., Wn,t )T
n- dimensional Wiener process or
Brownian motion with independent components
We propose
• Dynamics of short-rate is modeled on the P measure by a
cascade of n mean-reverting difusions
• Long-term rate θr ∈ R is constant
• Adjustment speeds κ speed at which factors return to their
long-term trend
15

The dynamics of the n− dimensional state vector Xt = (x1,t , . . . , xn,t )
T
is deﬁned as
Assumption 1 (Factors Structure)
x0,t is the long-term rate equal to θr constant, at which the process
reverts in mean. The diffusion cascade is deﬁned for the factors
dxj,t = κj (xj−1,t − xj,t )dt + σj dWj,t (1)
for all j ∈ {1, ..., n}, with κ1, ...κn, σ1, ..., σn strictly positive constants.
Adjustment speeds are strictly increasing κ1 < κ2 < ... < κn.
16

The dynamics of the n− dimensional state vector Xt = (x1,t , . . . , xn,t )
T
is defined as
Assumption 1 (Factors Structure)
x0,t is the long-term rate equal to θr constant, at which the process
reverts in mean. The diffusion cascade is defined for the factors
dxj,t = κj (xj−1,t − xj,t )dt + σj dWj,t (1)
for all j ∈ {1, ..., n}, with κ1, ...κn, σ1, ..., σn strictly positive constants.
Adjustment speeds are strictly increasing κ1 < κ2 < ... < κn.
• The first factor x1,t is an Ornstein-Uhlenbeck process with mean
reversion to θr .
• For each j ≥ 1 the jth
factor xj,t reverts in mean towards the
(j − 1)th
factor xj−1,t .
• The factors are less persistent when j increases
16

Assumption 2 (Short Rate)
The instantaneous interest rate coincides with the last factor: rt = xn,t
for all t 0.
17

Assumption 2 (Short Rate)
The instantaneous interest rate coincides with the last factor: rt = xn,t
for all t 0.
the states vector Xt = (x1,t , ..., xn,t )T
∈ Rn
satisﬁes:
dXt = κ(θ − Xt )dt + Σ1/2
dWt , (2)
Where θ = (θ, ..., θ)T
, Σ is a diagonal matrix with constant elements
σ2
t , ..., σ2
n, and κ is a matrix with diagonal elements κj,j = κj and ele-
ments outside the diagonal κj,j−1 = −κj .
17

Σ =








σ2
1 0 0 . . . 0
0 σ2
2 0 . . . 0
0 0 σ2
3 . . . 0
...
...
...
... 0
0 0 0 0 σ2
n








κ =








κ1 0 0 . . . 0
−κ2 κ2 0 . . . 0
0 −κ3 κ3 . . . 0
...
...
...
... 0
0 0 0 −κn κn








Each equation separates the factors according to their average rever-
sion rates, that is, how quickly, in the face of a shock, they return to
their steady or long-term state.
18

Representation of Short Rate Factors
Proposition (Factors representation for the short rate)
The instantaneous interest rate rt is a afﬁne function of the initial
state vector X0 and of integrals of the innovation driving each factor
between 0 and t.
rt = θr +
n
j=1
aj (t)(xj,0 − θr ) +
n
j=1
σj
ˆ t
0
aj (t − s)dWj,s (3)
19

Representation of Short Rate Factors
Proposition (Factors representation for the short rate)
The instantaneous interest rate rt is a afﬁne function of the initial
state vector X0 and of integrals of the innovation driving each factor
between 0 and t.
rt = θr +
n
j=1
aj (t)(xj,0 − θr ) +
n
j=1
σj
ˆ t
0
aj (t − s)dWj,s (3)
The long-term expectation of the short rate limt→inf EP
t (rt+τ ), is θr .
19

Response Functions
Each function aj (τ) is the product convolution of the exponential prob-
ability density functions
20

Response Functions
Proposition (Response Functions)
For all j < n the response functions aj (τ) are hump-shaped and their
maximum response horizons are monotonically decreasing with j.
The function satisﬁes the closed-form expression
aj (τ) =
n
i=j
αi,j κi e−κi τ
, with αi,j =
κj+1 . . . κn
κi
n
k=j,k=i (κk − κi )
(4)
20

Response Functions
Proposition (Response Functions)
For all j < n the response functions aj (τ) are hump-shaped and their
maximum response horizons are monotonically decreasing with j.
The function satisﬁes the closed-form expression
aj (τ) =
n
i=j
αi,j κi e−κi τ
, with αi,j =
κj+1 . . . κn
κi
n
k=j,k=i (κk − κi )
(4)
Moreover
0 ≤
n
j=1
aj (τ) ≤ 1
for all τ ≥ 0. Akkouchi (2018).
20

Risk Premia and Bond Prices
Suppose there are no arbitrage opportunities, then there exists a risk-
neutral measure pricing measure Q
Then the value in time t of a zero coupon bond, with a monetary unit
as par value and maturity t + τ is given by
P(Xt , τ) = EQ
t exp(−
ˆ t+τ
t
rsds)
21

P(Xt , τ) = EQ
t exp(−
ˆ t+τ
t
rsds)
Let γ denote the risk premia vector, which measures the amount (by
unit of volatility) by which the return of a risky asset is expected to
outperform the known return on a risk-free asset
21

P(Xt , τ) = EQ
t exp(−
ˆ t+τ
t
rsds)
Let γ denote the risk premia vector, which measures the amount (by
unit of volatility) by which the return of a risky asset is expected to
outperform the known return on a risk-free asset
Assumption 3 (Risk neutral measure)
The risk-neutral measure Q deﬁned by the following Radon-Nikdm
derivative
dQ
dP t
=
n
j=1
exp −
ˆ t
0
γj dWj,s −
1
2
ˆ t
0
γ2
j ds (5)
21

Using Girsanov’s theorem, the dynamics of the factors under Q are
dxj,t = γj σj dt + κj (xj−1,t − xj,t )dt + σj dWQ
j,t (6)
22

Using Girsanov’s theorem, the dynamics of the factors under Q are
dxj,t = γj σj dt + κj (xj−1,t − xj,t )dt + σj dWQ
j,t (6)
Therefore, the dynamics of the factors remain similar (afﬁne) under the
risk neutral measure. In a matrix way, we have:
dXt = κ(θQ
− Xt )dt + Σ1/2
dWQ
t , (7)
Where θQ
portray the long-run mean of the state vector under the risk
neutral measure and is given by:
θQ
r = θr −
γ1σ1
κ1
, θQ
2 = −
γ2σ2
κ2
, . . . , θQ
n = −
γnσn
κn
(8)
22

Proposition 3 (Zero Coupon Bond Price)
The price at date t of a zero-coupon bond with maturity τ is given by
a afﬁne model to Xt :
P(Xt , τ) = exp[−b(τ)T
Xt − c(τ)] (9)
The coeﬁcients b(τ) and c(τ) satisfy the system of ordinary
differential equations:
b (τ) = en − κT
b(τ) (10)
c (τ) = b(τ)T
κθQ
−
1
2
b(τ)T
Σb(τ) (11)
With initial conditions b(τ) = 0 y c(τ) = 0, where en denotes a vector
with the value one in the n− th position, and zero elsewere.
23

Closed Form
Proposition 4 (Valuation under constant premia)
The price loadings bj (τ) satisfy
bj (τ) =
ˆ τ
0
aj (τ )dτ =
n
i=j
αi,j (1 − e−κi τ
) (12)
The long-run levels of the state vector under the measure Q are
θQ
r = θr − γ1σ1/κ1
θQ
2 = θr − γ1σ1 − γ2σ2/κ2
...
θQ
n = θr −
n
i=1
γi σi /κi
24

The intercept c(τ) is given by
c(τ) = θr κ1
n
i=1
αi,1 τ −
1 − e−κi τ
κi
−
n
j=1
γj σj
n
i=j
αi,j τ −
1 − e−κi τ
κi
−
n
j=1
σ2
j
2
n
i=j
n
k=j
αi,j αk,j τ −
1 − e−κi τ
κi
−
1 − e−κk τ
κk
+
1 − e−(κi +κk )τ
κi + κk
25

The intercept c(τ) is given by
c(τ) = θr κ1
n
i=1
αi,1 τ −
1 − e−κi τ
κi
−
n
j=1
γj σj
n
i=j
αi,j τ −
1 − e−κi τ
κi
−
n
j=1
σ2
j
2
n
i=j
n
k=j
αi,j αk,j τ −
1 − e−κi τ
κi
−
1 − e−κk τ
κk
+
1 − e−(κi +κk )τ
κi + κk
The long-run yield y∞ = limτ→+∞[b(τ)T
Xt + c(τ)]/τ satisﬁes
y∞ = θr −
n
j=1
σ2
j
κ2
j
γj
σj
κj +
1
2
, (13)
25

Introduction
Base speciﬁcation
Data and Estimation
Empirical Results
Conclusions
26

Base speciﬁcation
Assumption 4 (Adjustment Speeds)
The sequence of adjustment speeds follows a geometric progression:
κj = kbj−1
, j = 1, 2, . . . , n. (14)
where b > 1 and k > 0 are constant real numbers.
27

Base speciﬁcation
Assumption 4 (Adjustment Speeds)
The sequence of adjustment speeds follows a geometric progression:
κj = kbj−1
, j = 1, 2, . . . , n. (14)
where b > 1 and k > 0 are constant real numbers.
Assumption 6 (Risk Premia)
Factors risk premia are identical: γj = γ for all j
Under the risk-adjusted measure, the long-term levels of the ﬁrst factor
are θQ
= θr − γσ/k.
27

Base speciﬁcation
Assumption 5 (Volatilities)
Standard Model Factors volatilities are identical: σj = σ > 0 for all j.
28

Base speciﬁcation
Assumption 5 (Volatilities)
Standard Model Factors volatilities are identical: σj = σ > 0 for all j.
Sigma Variant Model Factors volatilities follow a geometric
progression adjusted by a new parameter s
σj = σb(j−1)s
> 0, j = 1, . . . , n
• The standard model is speciﬁed by 5 parameters (k, b, σ, θ, γ)
with θQ
= θr − γσ/k.
• The SV model has 6, (k, b, σ, θ, γ, s), adding the s parameter in
volatility.
28

Introduction
Data and Estimation
Empirical Results
Conclusions
29

Data
Sample
• Sovereign zero coupon curve of Colombia with maturities of 1, 2,
4, 6, 7, 8, 10 and 15 years, extracted from Bloomberg.
• Data are weekly with cut on Friday, with the average rate
between bid and ask. Linear Interpolation.
• From September 23, 2005 to December 28, 2018.
• 704 weekly observations of each of the 8 time series, for a total
of 5632 observations.
30

Data
Sample
• Sovereign zero coupon curve of Colombia with maturities of 1, 2,
4, 6, 7, 8, 10 and 15 years, extracted from Bloomberg.
• Data are weekly with cut on Friday, with the average rate
between bid and ask. Linear Interpolation.
• From September 23, 2005 to December 28, 2018.
• 704 weekly observations of each of the 8 time series, for a total
of 5632 observations.
Robustness, two subsamples
1. Estimation: from September 23, 2005 to December 30, 2016, for
a total of 589 weekly observations for each time series.
2. Forecast: from January 6, 2017 to December 28, 2018, with a
total of 115 observations.
30

Table 1: Yield Rate Statistics Summary
Autocorrelation
Maturity Mean Desv. Std Skew. Kurt. 1
1 ao 5.695 1.849 1.007 -0.037 0.9937
2 aos 6.201 1.799 1.126 0.239 0.9958
4 aos 6.859 1.770 0.909 0.137 0.9904
6 aos 7.331 1.721 0.823 0.112 0.9964
7 aos 7.607 1.525 0.907 0.418 0.9975
8 aos 7.567 1.665 0.448 -0.752 0.9982
10 aos 7.776 1.599 0.807 0.112 0.9959
15 aos 7.953 1.429 0.681 0.029 0.9938
Average 7.077 1.669 0.8385 0.032 0.9952
Table 1 reports a statistical summary of weekly observations (with Friday mid-
quotes) of yield rates with maturities of 1,2,4,6,7,8,10 and 15 years. Each se-
ries contains 704 weekly observations from September 23, 2015 to December
28, 2018. Entries report: mean (Mean), standard deviation (Dev. Std), skewness
(Skew), excess kurtosis (Kurt) and ﬁrst order weekly autocorrelation.
31

Space-State Model
• It’s assumed that bond prices are observed in time intervals ∆t.
• The yield rates relate the prices of the zero coupon bonds
through a logarithmic transformation of the function (9), obtaining
a measurement equation:
y(t, τ) =
b(τ)T
Xt + c(τ)
τ
+ et (15)
32

Space-State Model
• It’s assumed that bond prices are observed in time intervals ∆t.
• The yield rates relate the prices of the zero coupon bonds
through a logarithmic transformation of the function (9), obtaining
a measurement equation:
y(t, τ) =
b(τ)T
Xt + c(τ)
τ
+ et (15)
• The state propagation equation (or transition equation) is:
Xt = A + ΦXt−∆t + ΣX εt , (16)
Where ∆t = 1/52, Φ = exp(−κ∆t), In denotes an identity matrix
n -dimensional, A = (In − Φ) θ, {εt } is i.i.d N(0, In),
With:
• ΣX = σ2
∆tIn for the standard model.
• ΣX = σ2
j ∆tIn con σ2
j = σ2
b(j−1)s
for the SV model.
32

Estimation and Optimization
• Being a model with unknown factors and parameters, and
possessing the desirable conditions of linearity and Gaussian
behavior is estimated through a Kalman ﬁlter.
• Initial values are assumed ˆx1|0 y P1|0 where ˆx1|0 = θr , the long
term rate y P1|0 = σ2
2k I.
33

Estimation and Optimization
• Being a model with unknown factors and parameters, and
possessing the desirable conditions of linearity and Gaussian
behavior is estimated through a Kalman ﬁlter.
• Initial values are assumed ˆx1|0 y P1|0 where ˆx1|0 = θr , the long
term rate y P1|0 = σ2
2k I.
• Algorithm of Broyden Fletcher Goldfarb Shanno with limited
memory and boundaries or L-BFGS-B, optimizes the likelihood
function:
log L = −
NT
2
log(2π) −
1
2
T
t=1
det |Σe| −
1
2
T
t=1
et [Σe]
−1
et (17)
Where Σe is the covariance matrix of the measurement error and
et the estimation error of the measurement equation.
33

Figure 1: Estimation and Optimization Algorithm
Initial values
κ, 𝑏, 𝜎, 𝜃𝑟, γ, 𝜎𝑒
Maximum Likelihood for T obs.
Minimize the log likelihood func.
negative sum conditional to the values
generate by the filter.
Kalman Filter for n factors on T
observations.
Estimation: ො𝑥 𝑡, Σ 𝑒, 𝑒𝑡
Optimize L-BFGS-B. The function
value reaches its minimum?
Finish optimization. Values that
minimize the log likelihood negative sum
have been found.
Select new values
κ, 𝑏, 𝜎, 𝜃𝑟, γ, 𝜎𝑒
No
Yes
34

Introduction
Data and Estimation
Empirical Results
Conclusions
35

In sample Estimation
Table 2: Estimated Parameters and Standard Errors bp
Standard Model
Parameters
n k b σ θ γ σ2+
e θQr
L
1 1.0572*** 0.0110 0.0702* -0.4315*** 0.0311*** 0.0702 -12148.49
(0.0233) (—-) (0.1076) (0.0412) (0.0023) (0.0044)
2 0.4977*** 1.4088*** 0.0312*** 0.0753*** -0.4341*** 0.03741*** 0.0761 -12120.65
(0.0029) (0.0028) (0.0053) (0.0110) (0.00001) (0.0050)
3 0.5795*** 1.4043*** 0.0106*** 0.0455*** -0.4317*** 0.0302*** 0.0456 -12755.38
(0.0011) (0.0010) (0.0007) (0.0013) (0.000005) (0.0026)
4 1.7900*** 1.4053*** 0.0305 0.0610* -0.4351*** 0.0612 0.3164 -13107.42
(0.1007) (0.0485) (0.0548) (0.0336) (0.0129) (0.7079)
5 1.3255*** 2.0069*** 0.02314 0.0644 -0.4345*** 0.0249*** 0.0646 -12796.83
(0.0909) (0.1139) (0.0343) (0.0721) (0.0005) (0.3373)
6 0.1975*** 1.1218*** 0.0187*** 0.0466 0.1290*** 0.0739 0.0464 -9287.79
(0.0522) (0.0442) (0.0476) (0.6501) (0.0004) (6.4866)
7 0.3771*** 1.8281** 0.0151 0.0572 -0.4105*** 0.0099 0.0574 -14163.49
(0.0798) (0.7829) (0.0211) (0.0973) (0.0025) (0.483)
8 0.4547*** 1.6104*** 0.0126*** 0.0379*** -0.4309*** 0.0370*** 0.0381 -11532.20
(0.0031) (0.0035) (0.0025) (0.0034) (0.00002) (0.0098)
Table 2 reports the maximum likelihood estimates and standard errors (in parentheses) of the standard model parameters presented
in Calvet (2018). Each row represents a set of parameters estimated with the cascade with n = 1 to n = 8 components. The L
column reports the maximum weekly average likelihood log value for each model. Asymptotically, the statistician has a standard
normal distribution. Signiﬁcance level: * 10 %, ** 5 % and *** 1 %.
+ Both measurement errors and their deviations are found in bp.
36

Table 3: Estimated Parameters and Standard Errors bp
SV Model
Parameters
n k b σ θ γ σ2+
e s θQ
r L
1 0.2596*** 0.0257 0.0601*** -0.1406*** 0.7948*** 0.0605 -6862.94
(0.0256) (—-) (0.0253) (0.0118) (0.0012) (0.0030) (—-)
2 0.1234*** 1.1523*** 0.0122*** 0.0286 0.4575*** 0.3780*** 0.1890*** 0.0280 -9876.86
(0.0007) (.0001) (0.0019) (0.1924) (0.00002) (0.0025) (0.00001)
3 1.0768*** 2.3100*** 0.0261*** 0.0317*** -0.4314*** 0.1110*** 0.4989*** 0.0314 -10508.42
(0.0069) (0.0040) (0.0066) (0.0055) (0.00007) (0.0023) (0.0041)
4 0.3271*** 1.4317*** 0.0024 0.0805*** -0.4463*** 0.0130*** 0.5027*** 0.0805 -15171.54
(0.0202) (0.1635) (0.1041) (0.0155) (0.0577) (0.0075) (0.0519)
5 0.3373*** 1.4027*** 0.0108*** 0.0526* -0.4346*** 0.0103 0.5007*** 0.0528 -13998.85
(0.0466) (0.0601) (0.0057) (0.0307) (0.0494) (0.0750) (0.0770)
6 1.4325*** 1.3914*** 0.0441 0.0302 -0.4326*** 0.0099 0.5094*** 0.0309 -14192.72
(0.0827) (0.0571) (0.0271) (0.0245) (0.0787) (0.7834) (0.1013)
7 0.1422*** 1.1927*** 0.0014 0.0963*** 0.2779*** 0.0046 0.4909*** 0.0963 -15587.02
(0.0016) (0.0009) (0.0038) (0.0001) (0.00001) (0.2547) (0.0002)
8 0.2492*** 1.2806*** 0.0291* 0.0647** -0.1018*** 0.0094 0.3817*** 0.0650 -13291.66
(0.0060) (0.0267) (0.0152) (0.0274) (0.0306) (1.5207) (0.0215)
Table 3 reports the maximum likelihood estimates and standard errors (in parentheses) of the variant model sigmaj parameters. Each row
represents a set of parameters estimated with the cascade with n = 1 to n = 8 components, where the measurement error is in basic points.
The cL column reports the maximum weekly average likelihood log value for each model. Asymptotically, the statistician has a standard normal
distribution. Signiﬁcance level: * 10 %, ** 5 % and *** 1 %.
+ Both measurement errors and their deviations are found at bp.
37

In sample Fit
• The Root Mean Square Error (RMSE)
• Determine how far the values generated by the real model move
away.
• The RMSE has the following formula:
RMSE = E yt,τ − ˆyt,τ yt,τ − ˆyt,τ (18)
38

Table 4: RMSE in bp Standard Model
n
Maturity 1 2 3 4 5 6 7 8
1 ao 0.5037 0.8222 0.6478 0.6681 0.4894 2.7089 0.9226 1.3220
2 aos 0.4569 0.6496 0.3949 0.2561 0.1153 2.5289 0.7779 1.0354
4 aos 0.4260 0.2942 0.0763 0.1182 0.1850 2.2392 0.5338 0.5584
6 aos 0.4245 0.0864 0.1884 0.2196 0.2692 1.9981 0.3657 0.2788
7 aos 0.3843 0.0955 0.2585 0.2126 0.2545 1.8555 0.3460 0.2165
8 aos 0.4140 0.1902 0.3140 0.2680 0.3036 1.7671 0.3029 0.2039
10 aos 0.4125 0.2751 0.3747 0.2756 0.3093 1.5476 0.1834 0.14283
15 aos 0.3907 0.4614 0.4546 0.2840 0.3091 1.0422 0.1572 0.2479
Average 0.4266 0.3593 0.3384 0.2878 0.2794 1.9609 0.4487 0.5014
Table 5: RMSE in bp SV Model
n
Maturity 1 2 3 4 5 6 7 8
1 ao 0.2778 0.7127 0.1975 0.3307 0.6080 0.6390 0.7046 0.4186
2 aos 0.2212 0.4690 0.1971 0.2139 0.5500 0.5233 0.5642 0.3800
4 aos 0.2411 0.1197 0.4391 0.0661 0.4355 0.2320 0.3726 0.3575
6 aos 0.2971 0.2862 0.5649 0.1109 0.3486 0.1797 0.2875 0.3765
7 aos 0.3047 0.4390 0.6081 0.1057 0.3566 0.1483 0.2548 0.4222
8 aos 0.3363 0.5540 0.6375 0.1662 0.3007 0.2135 0.2408 0.4062
10 aos 0.3549 0.7759 0.6786 0.1678 0.2154 0.2002 0.1837 0.3558
15 aos 0.3717 1.2278 0.7155 0.2016 0.1918 0.2241 0.1502 0.2936
Average 0.3006 0.5730 0.5048 0.1704 0.3758 0.2950 0.3448 0.3762 39

Introduction
Data and Estimation
Empirical Results
Conclusions
40

Conclusions
• Estimated values for different dimensions ﬂuctuate along the size
of the cascade, for both models. Possible causes: Linearization
of measurement equation, data interpolation, optimization.
41

Conclusions
• The standard model is more stable than the SV model, but the
latter is more consistent in results in the sample and out-of-the
sample (work in progress).
41

Conclusions
• For this type of afﬁne model, it is advisable to use at least 3 factors
for the adjustment in the term structure data in Colombia
41

Conclusions
• For this type of afﬁne model, it is advisable to use at least 3 factors
for the adjustment in the term structure data in Colombia
• Dimension-invariant afﬁne models do not seem suitable for fore-
casting the dynamics of Colombian TS (work in progress).
However, can be readily applied to pricing, bond portfolio risk
management and risk premium analysis.
41

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