stochastic modeling of yield curve shifts using functional...
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Stochastic modeling of yield curve shifts using functional data analysis
R A S M U S R E H N
Master of Science Thesis Stockholm, Sweden 2014
Stochastic modeling of yield curve shifts using functional data analysis
R A S M U S R E H N
Master’s Thesis in Mathematical Statistics (30 ECTS credits) Master Programme in Mathematics (120 credits) Royal Institute of Technology year 2014 Supervisor at KTH was Boualem Djehiche Examiner was Boualem Djehiche
TRITA-MAT-E 2014:38 ISRN-KTH/MAT/E--14/38--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci
Stochastic modeling of yield curve shifts using
functional data analysis
Rasmus Rehn
Royal Institute of Technology
June 16, 2014
Abstract
This thesis approaches the problem of modeling the multivariate distribution
of interest rates by implementing a novel tool of statistics known as functional
data analysis (FDA). This is done by viewing yield curve shifts as distinct but
continuous stochastic objects defined over a continuum of maturities. Based on
these techniques, we provide two stochastic models with different assumptions
regarding the temporal dependence of yield curve shifts and compare their per-
formance with empirical data. The study finds that both models replicate the
distributions of yield changes with medium- and long-term maturities, whereas
none of the models perform satisfactory at the short segment of the yield curve.
Both models, however, appear to accurately capture the cross-sectional depen-
dence.
Acknowledgements
I would like to express my great appreciation to my tutor Boualem Djehiche
at Royal Institute of Technology for introducing me to functional data analysis
as a modeling tool for finance and risk management. Also, I would like to
thank Jonas Nilsson, Fredrik Bohlin, Tomas Hirsch, Ulrika Trolle and Cecilia
Pettersson for giving me the opportunity to write this thesis at Handelsbanken
Capital Markets and for providing me with the data required for the study.
1
Contents
1 Introduction 3
2 Previous research 4
3 Data 5
3.1 Yield curve data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Construction of yield curve shifts . . . . . . . . . . . . . . . . . . . . 6
3.3 Construction of continuous yield curve shifts . . . . . . . . . . . . . 10
4 Methodology 11
4.1 Functional data analysis . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Functional principal component analysis . . . . . . . . . . . . . . . . 14
4.3 Functional autoregression estimation . . . . . . . . . . . . . . . . . . 16
4.4 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.5 Methods for evaluating model performance . . . . . . . . . . . . . . 20
5 Results 21
5.1 Empirical functional principal components . . . . . . . . . . . . . . . 21
5.2 FAR(1) predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Simulation model performance . . . . . . . . . . . . . . . . . . . . . 27
6 Conclusion 33
References 34
2
1 Introduction
The value of a bond portfolio depends on yield levels of its included maturities. An
investor holding such a portfolio is therefore exposed to interest rate risk which is
simply the uncertainty regarding the future portfolio value due to stochastic changes
in yields. To fully understand this risk exposure, one needs to evaluate the joint prob-
ability distribution of yield changes over given time intervals. However, determining
this distribution function might be difficult. Doing so requires an accurate under-
standing of how multiple stochastic variables are distributed and how they depend
on each other.
As a result, risk analysts often use simplifications and approximations when analyz-
ing interest rate risk of portfolios. Usually, this is done by considering how much
portfolio values change if the yield curve shifts in parallel by a certain amount. Al-
though such methods might give useful indications of interest rate sensitivity, they
are still unrepresentative of true value fluctuations. The parallel shift assumption
might be sufficient for adjacent maturities or over small time intervals, but does not
realistically describe a general yield curve transformation. Most importantly, the
parallel shift assumption does not give any information about the probability dis-
tribution of the future portfolio value which is what a rational investor cares about
ultimately.
We approach this problem by modeling yield curve shifts as distinct but continuous
stochastic objects defined over a continuum of maturities rather than multivariate
stochastic variables defined for a finite number of maturities. The framework un-
derlying this technique is known as functional data analysis (FDA) which is a novel
field of statistics dating back to the early 1990’s [9]. It however relies on findings
– mostly the theory of Hilbert spaces and functional analysis – which are far older
branches of mathematics.
Based on this framework, we provide two stochastic models which attempt to de-
scribe realistic yield curve shifts. The first of these models assumes that yield curve
3
shifts are independent across time whereas the second model incorporates temporal
dependence through the introduction of the functional autoregressive (FAR) opera-
tor. Both models are found to replicate the marginal distributions of yield changes
for medium-term and long-term maturities, whereas distributions for the shorter
segment of the yield curve appear less well-captured. Still, both models appear
to resemble much of the cross-sectional dependence inherent in the yield curve, as
measured by sample correlation estimates.
The remainder of this thesis is organized as follows. Section 2 provides an overview
of existing yield curve modeling literature. Thereafter, Section 3 describes the yield
curve data used in the study and the main steps of the construction of continuous
yield curve shifts. Furthermore, Section 4 outlines the mathematical framework
underlying FDA, functional principal component analysis as well as the theory of
FAR operators. Finally, Section 5 describes the main empirical results whereas
Section 6 concludes.
2 Previous research
The literature on yield curve modeling has grown tremendously over the last couple
of decades. Existing proposals can be broadly described in terms of three categories.
A first set of yield curve models is based on modeling the instantaneous short rate
which is the theoretical interest rate prevailing over an infinitesimal time interval
[t, t+dt]. An early work in this field is provided by Vasicek [14] who models the short
rate as a mean-reverting stochastic process and derives a corresponding yield curve
model. Despite analytical tractability of this model, its validity has been questioned
partly due to its inability to reproduce strictly non-negative interest rates. Cox,
Ingersoll and Ross [2] further add to this class of models by modeling the variance of
the stochastic short-rate process as a function of the short rate itself, such that non-
negative interest rates are ensured. Other important contributions in the short-rate
class of models include Ho and Lee [6] and Hull and White [8].
4
A drawback of these models, however, resides within their inability to model yield
curves such that arbitrage opportunities can be excluded. A second branch of the
literature, generalized by Heath, Jarrow and Morton [5], therefore proposes models
which are consistent with no-arbitrage restrictions. This is done by modeling the
instantaneous forward curve, rather than the instantaneous short rate, such that
arbitrage opportunities can be ruled out.
A third category of yield curve literature aims at modeling yield curve shapes by
fitting suitable basis functions to observed yield curves. An early piece of work
in this class is provided by Nelson and Siegel [11] who propose a parsimonious
three-factor model that tends to describe the yield curve rather efficiently. A four-
factor extension of this model has been proposed by Svensson [13]. Furthermore,
Diebold and Li [3] propose a dynamic factor model based on the Nelson and Siegel
[11] framework. A similar contribution is provided by Hays et al [4]. This model
however uses the theory of FDA which improves the accuracy of the functional form
used to express the yield curve function.
This paper adds to existing yield curve literature by expanding the use of FDA tech-
niques in yield curve modeling. While Hays et al [4] mainly study the predictability
performance of their model, this paper rather focuses on the distribution of yield
curve shifts from a risk management perspective.
3 Data
3.1 Yield curve data
In order to evaluate the applicability of FDA methods (described in detail in Sec-
tion 4) we apply these on empirical observations of German sovereign bond yields.
The data set spans weekly observations between November 18th, 2002 and March
10th, 2014, equivalent to 590 consecutive observations. At each date, we use the
mid values between quoted closing ask and bid yields of 14 different maturities re-
spectively between 3 months and 30 years. All observations are downloaded from
5
the Bloomberg platform and the full data set is illustrated in Figure 1.
May01 Feb04 Nov06 Aug09 May12 Feb15 0
10
20
30
−1
0
1
2
3
4
5
6
Maturity [years]
Yie
ld [%
]
Figure 1: German sovereign yield curves in the period November 18th, 2002 - March 10th, 2014,corresponding to 590 consecutive observations.
3.2 Construction of yield curve shifts
Let yn(τ0), yn(τ1), ..., yn(τm) denote the set of m+ 1 observed yields with maturities
τ0, τ1, ..., τm at each date n = 1, 2, ..., N , where τ0 < τ1 < ... < τm. Furthermore, let
xn(τi) = yn(τi)− yn−1(τi) (1)
denote the change in the yield with maturity τi between week n−1 and n. Ultimately,
we are interested in understanding the joint behavior of xn(τ0), xn(τ1), ..., xn(τm).
However, the FDA techniques described in Section 4 requires yield curve shifts to
be defined on the interval [0, 1]. To enforce this, each maturity is transformed into
the dimensionless quantity
ti =τi − τ0τm
∈ [0, 1] (2)
6
and the remainder of the paper will analyze the properties ofXn(t0), Xn(t1), ..., Xn(tm),
where
Xn(ti) := xn(τi). (3)
A summary of the maturities τi included in the data set and their transformed
values ti is provided in Table 1. Furthermore, estimated cross-sectional correlations
of (Xn(t0), Xn(t1), . . . , Xn(tm)) are provided in Table 2 whereas Figure 2 illustrates
histograms of yield changes across all 14 maturities.
Maturity Abbreviation i τi ti3 months 3M 1 0.25 0.006 months 6M 2 0.5 0.011 year 1Y 3 1 0.032 years 2Y 4 2 0.063 years 3Y 5 3 0.094 years 4Y 6 4 0.135 years 5Y 7 5 0.166 years 6Y 8 6 0.197 years 7Y 9 7 0.238 years 8Y 10 8 0.269 years 9Y 11 9 0.2910 years 10Y 12 10 0.3320 years 20Y 13 20 0.6630 years 30Y 14 30 1.00
Table 1: Summary of observed maturities τi along with transformed values ti = τi−τ1τm
∈ [0, 1].
7
−2
02
050100
150
200
03M
−2
02
020406080100
120
140
06M
−2
02
01020304050607001
Y
−2
02
0510152025303502
Y
−2
02
0510152025303503
Y
−2
02
051015202504
Y
−2
02
051015202505
Y
−2
02
051015202506
Y
−2
02
05101520253007
Y
−2
02
05101520253008
Y
−2
02
051015202509
Y
−2
02
051015202510
Y
−2
02
05101520253020
Y
−2
02
051015202530
Y
Fig
ure
2:H
isto
gra
ms
of
obse
rved
wee
kly
yie
ldch
anges
for
14
diff
eren
tm
atu
riti
esin
the
range
3m
onth
s-30
yea
rs.
8
3M6M
1Y2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
20Y
30Y
3M1.
000.
660.
430.
22
0.2
00.1
70.1
50.1
30.1
20.1
30.1
10.1
30.1
40.1
36M
0.66
1.00
0.74
0.46
0.4
00.3
70.3
40.3
20.3
00.2
80.2
50.2
60.2
60.2
51Y
0.43
0.74
1.00
0.80
0.7
20.6
80.6
40.6
00.5
60.5
30.4
90.4
80.3
90.3
62Y
0.22
0.46
0.80
1.00
0.9
30.9
10.8
80.8
40.8
10.7
70.7
40.7
20.6
00.5
53Y
0.20
0.40
0.72
0.93
1.0
00.9
80.9
40.8
90.8
60.8
30.8
00.7
90.6
80.6
34Y
0.17
0.37
0.68
0.91
0.9
81.0
00.9
80.9
40.9
30.9
10.8
80.8
70.7
60.7
15Y
0.15
0.34
0.64
0.88
0.9
40.9
81.0
00.9
70.9
60.9
40.9
20.9
10.8
10.7
66Y
0.13
0.32
0.60
0.84
0.8
90.9
40.9
71.0
00.9
90.9
80.9
60.9
50.8
50.8
07Y
0.12
0.30
0.56
0.81
0.8
60.9
30.9
60.9
91.0
00.9
90.9
80.9
70.8
80.8
38Y
0.13
0.28
0.53
0.77
0.8
30.9
10.9
40.9
80.9
91.0
00.9
90.9
80.9
00.8
69Y
0.11
0.25
0.49
0.74
0.8
00.8
80.9
20.9
60.9
80.9
91.0
00.9
90.9
20.8
810
Y0.
130.
260.
480.
72
0.7
90.8
70.9
10.9
50.9
70.9
80.9
91.0
00.9
30.9
020
Y0.
140.
260.
390.
60
0.6
80.7
60.8
10.8
50.8
80.9
00.9
20.9
31.0
00.9
830
Y0.
130.
250.
360.
55
0.6
30.7
10.7
60.8
00.8
30.8
60.8
80.9
00.9
81.0
0
Tab
le2:
Est
imate
dcr
oss
-sec
tional
corr
elati
on
coeffi
cien
tsfo
robse
rved
wee
kly
yie
ldch
anges
.
9
3.3 Construction of continuous yield curve shifts
As outlined in Section 1, FDA enables the risk analyst to model yield curve shifts
as continuous stochastic functions. Therefore, before applying FDA methods, the
observed (discrete) data needs to be interpolated such that continuous objects are
obtained. This is done by fitting the observations Xn(ti) to natural spline functions
Si,n(t) for i = 0, 1, ...,m − 1, where t (without subscript) refers to the continuous
maturity variable on the interval [0, 1].
For each observation date n and for each sub-interval [ti, ti+1], i = 0, 1, ...,m we fit
the third degree polynomial Si,n(t) = ai,n + bi,n(t − ti) + ci,n(t − ti)2 + di,n(t − ti)3
such that the conditions
Si,n(ti) = Xn(ti), i = 0, 1, ...,m− 1
Si,n(ti+1) = Xn(ti+1), i = 0, 1, ...,m− 2
Si+1,n(ti+1) = Si,n(ti+1), i = 0, 1, ...,m− 2
S′i+1,n(ti+1) = S′i,n(ti+1), i = 0, 1, ...,m− 2
S′′i+1,n(ti+1) = S′′i,n(ti+1), i = 0, 1, ...,m− 2
S′′i,n(t0) = S′′i,n(tm) = 0
(4)
are fulfilled. The interpolated continuous yield curve shifts are then given by
Xn(t) =
S0,n(t), t ∈ [t0, t1]
S1,n(t), t ∈ [t1, t2]
...
Sm−1,n(t), t ∈ [tm−1, tm].
(5)
The first two spline conditions in Eq. (4) ensure that the fitted splines pass through
the observed discrete points Xn(ti). Furthermore, the third condition guarantees
continuity in Xn(t) across the full interval [0, 1], whereas the fourth and fifth con-
ditions generate smooth shapes through enforcement of continuous first and second
10
derivatives. Finally, the last two conditions are boundary conditions which have
limited effects on the overall shape of the functions.
In order to determine the coefficients ai,n, bi,n, ci,n, di,n for i = 0, 1, ...,m and n =
1, 2, ..., N Burden and Faires [1] propose the following steps:
1. Define αi,n = 3hi
(Xn(ti+1) − Xn(t1)) − 3hi−1
(Xn(ti) − Xn(ti−1)), where hi =
ti+1 − ti.
2. Define the variables li, µi and zi,n such that li = 2(ti+1 − ti−1) − hi−1µi−1,
µi = hi/li and zi,n =αi,n−hizi−1,n
lifor i = 1, 2, ...,m − 1. Furthermore, set
l0 = lm = 1 and z0,n = zm,n = 0 as well as cn,m = 0.
3. The polynomial coefficients are given as ai,n = ti, bi,n =ai+1,n−ai,n
hi−hi ci+1,n+2ci,n
3 ,
ci,n = zi,n − µici+1,n and di,n =ci+1,n−ci,n
3hiwhich are obtained by solving the
corresponding linear system of equations.
Interpolated functions Xn(t) satisfying the spline conditions are illustrated in Figure
3. The remaining analysis will mainly focus on these observations and the methods
for doing this are described in the next section.
4 Methodology
4.1 Functional data analysis
This section provides an overview of the mathematical framework underlying FDA.
The theory described here is based on the book by Horvath and Kokoszka [7] and
it relies on the mathematics of Hilbert spaces. In particular, the analysis will con-
sider the space L2 = L2([0, 1]) which is the set of measurable real-valued functions
u = {u(t), t ∈ [0, 1]} with the square integrable property∫ 10 u
2(t)dt < ∞. L2 is a
separable Hilbert space with the inner product
〈u, v〉 =
∫ 1
0u(t)v(t)dt (6)
11
0 0.2 0.4 0.6 0.8 1−1.5
−1
−0.5
0
0.5
1
t
Xn(t
)
Figure 3: Continuous yield curve shifts.
and the norm
‖u‖2 = 〈u, u〉 =
∫ 1
0u2(t)dt. (7)
Furthermore, the analysis makes use of operators in L which is the space of bounded
linear operators on L2.
Throughout the remainder of the paper, the continuous yield curve shifts Xn(t),
n = 1, 2, ..., N will be viewed as realizations of the stochastic function X = {X(t), t ∈
[0, 1]} which is a random element in L2, equipped with the Borel σ-algebra. Fur-
thermore, it will be assumed that X is square integrable in the sense that
E‖X‖2 = E[〈X,X〉] = E
[∫ 1
0X2(t)dt
]<∞. (8)
Provided that this assumption holds, there is a unique function in L2
µ(t) = E[X(t)] (9)
12
which is known as the mean function of X. Furthermore, conditional on the square
integrable Condition (8), the covariance operator of X is defined as
C(u) = E[〈(X − µ) , u〉 (X − µ)] ∈ L (10)
for u ∈ L2. It holds that
C(u) =
∫ 1
0c(t, s)u(s)ds (11)
where
c(t, s) = E[(X(t)− µ(t)) (X(s)− µ(s))] (12)
is known as the covariance function. The sample counterparts of the mean function,
covariance operator and covariance functions are defined respectively as
µ(t) =1
N
N∑j=1
Xj(t), (13)
C(u) =1
N
N∑j=1
〈Xj − µ, u〉 (Xj − µ) (14)
and
c(t, s) =1
N
N∑j=1
(Xj(t)− µ(t)) (Xj(s)− µ(s)) , (15)
where u ∈ L2.
In the remaining analysis, it will be assumed that µ(t) = 0 which is supported by
the economic intuition that yields are drift-less in the sense that they tend to remain
within certain bounds over time. This assumption is verified by checking the sample
mean function µ(t) for the observed data set. Computations show that this function
is close to zero for all t ∈ [0, 1].
Provided that E‖X‖4 < ∞, which we have implicitly assumed by Condition (8),
the covariance operator is a symmetric positive-definite so called Hilbert-Schmidt
13
operator in L. As such, it can be decomposed as
C(u) =∞∑k=1
λk〈u, vk〉vk (16)
where vk and λk are the eigenfunctions and eigenvalues of C respectively, such that
C(vk) = λkvk. (17)
The eigenfunctions (also known as principal components) form an orthogonal basis
in L2 which allows the yield curve shifts X(t) to be expressed as
X(t) =∞∑k=1
ξkvk (18)
where the ξk = 〈X, vk〉 are known as the principal component scores of X. This
expansion is known as the Karhuen-Loeve expansion and provides the foundation of
functional principal component analysis (FPCA) described next.
4.2 Functional principal component analysis
In the context of modeling yield curve shifts, we are interested in expressing these
functions through an efficient basis expansion in L2. Ideally, we want this expan-
sion to capture as much as possible of the characteristics inherent in the functions
with as few basis functions as possible. For this purpose, let u1, u2, ..., up, denote
the basis functions of an orthonormal basis. We want to choose this basis such
that the observed yield curve shifts can be closely approximated by the expansion∑pk=1〈X,uk〉uk for a given value of p. This problem is equivalent to that of mini-
mizing
S2 =N∑n=1
‖Xi −p∑
k=1
〈Xn, uk〉uk‖2. (19)
Horvath and Kokoszka [7] show that this expression is minimized if u1 = v1, u2 =
v2, ..., up = vp, where v1, v2, ..., vp are the principal components corresponding to the
14
p largest eigenvalues λ1 > λ2 > ... > λp of the empirical covariance operator C.
The basis v1, v2, ..., vp can hence be viewed as the optimal orthonormal basis with
which we can describe the data. In particular, this result suggests that the Karhuen-
Loeve expansion in Eq. (18) can be well approximated by its finite counterpart
X(t) =
p∑k=1
ξkvk. (20)
For a given choice of p, this basis captures as much as possible of the observed
variance. To see why this is the case, consider the statistic
1
N
N∑n=1
〈Xn, u〉2 = 〈C(u), u〉 (21)
which can be interpreted as the sample variance along the direction of the function
u. Since v1, v2, ..., vN constitutes an orthogonal basis, it holds that
1
N
N∑n=1
‖Xn‖2 =1
N
N∑n=1
N∑q=1
〈Xn, vq〉2 =N∑q=1
1
N
N∑n=1
〈Xn, vq〉2 =N∑q=1
λq. (22)
Hence, the principal component vk explains a fraction of the total empirical variance
equivalent to λk∑Nq=1 λq
. Since λ1 > λ2 > ... > λp, the basis v1, v2, ..., vp therefore
captures the highest possible fraction of the variance for a given choice of p.
In order to compute this basis numerically, we follow the procedure suggested by
Ramsay and Silverman [12]. Let s1, s2, ..., sT denote discretized values of t. Further-
more, let
X =
X1(s1) X1(s2) . . . X1(sT )
X2(s1) X2(s2) . . . X2(sT )...
......
XN (s1) XN (s2) . . . XN (sT )
. (23)
The discretized sample covariance matrix can then be defined as
C =1
NX′X. (24)
15
Furthermore, let uk and ρk denote the eigenvector and eigenvalues of C such that
Cuk = ρkuk. An approximate discrete form of the empirical functional eigenequation
C(vk) = λkvk (25)
is then given by1
TCvk = λvk (26)
where the discretized eigenfunctions can be calculated as
vk =√Tuk (27)
provided that uk is a normalized eigenvector of C. In particular, vk contains the
elements vk(s1), vk(s2), ..., vk(sT ) which approximately replicate the true functions
vk(t) if the discretization is sufficiently dense.
4.3 Functional autoregression estimation
This section describes the theory of FAR processes and the derivation of an empirical
estimate of the FAR(1) operator as suggested by Horvath and Kokoszka [7]. Let
{Xn,∞ < n <∞} be a sequence of random yield curve shifts in L2. These are said
to follow a FAR(1) model if
Xn(t) = Ψ(Xn−1) + εn(t) (28)
where Ψ ∈ L is the FAR(1) operator and {εn,∞ < n < ∞} is a sequence of mean
zero iid errors in L2 such that E‖εn‖2 <∞.
In order to find an empirical estimator of Ψ, let
C1(u) = E[〈Xn, u〉Xn+1] (29)
denote the lag-1 operator which can be expressed in terms of the covariance operator
16
as C1 = ΨC. It follows that the FAR(1) operator can be obtained as
Ψ = C1C−1 (30)
provided that the inverse of the covariance operator C−1 exists. The representation
in Eq. (16) suggests that this may be expressed as
C−1(u) =
∞∑k=1
λ−1k 〈u, vk〉vk. (31)
However since ‖C−1(vk)‖ = λ−1k → ∞ as k → ∞ it is unbounded and therefore
badly suited for estimation. It is only defined in the subspace spanned by the first
q principal components such that λ1 ≥ λ2 ≥ ... ≥ λq > λq+1 = 0. Therefore, the
inverse of the empirical covariance operator may be estimated with the first p ≤ q
principal components as
ˆIC(u)p =
p∑k=1
λ−1k 〈u, vk〉vk. (32)
This calculation becomes a trade-off between choosing p small enough to avoid re-
ciprocals of small eigenvalues yet large enough to resemble as much as possible of
the information inherent in the data set.
An empirical estimator of C1 is given by
C1(u) =1
N − 1
N∑n=1
〈Xn, u〉Xn+1. (33)
Along with Eq. (30) and Eq. (32), this suggests that an empirical estimator of Ψ
17
can be expressed as
Ψp(u) = C1ˆICp(u) = C1
(p∑
k=1
λ−1k 〈u, vk〉vk
)=
=1
N − 1
N−1∑n=1
⟨Xn,
p∑k=1
λ−1k 〈u, vk〉vk
⟩Xn+1 =
=1
N − 1
N−1∑n=1
p∑k=1
λ−1k 〈u, vk〉〈Xn, vk〉Xn+1.
(34)
In addition, Horvath and Kokoszka [7] suggest smoothing the operator by using
the approximation implied by Eq. (20) such that Xn+1 ≈∑p
i=1〈Xn+1, vi〉vi. The
empirical operator then becomes
Ψp(u) =1
N − 1
N−1∑n=1
p∑k=1
p∑i=1
λ−1k 〈u, vk〉〈Xn, vk〉〈Xn+1, vi〉vi. (35)
The estimator given in Eq. (35) is a kernel operator in the sense that
Ψp(u) =
∫ 1
0ψp(t, s)u(s)ds (36)
where the kernel can be expressed as
ψp(t, s) =1
N − 1
N−1∑n=1
p∑k=1
p∑j=1
λ−1k 〈Xn, vk〉〈Xn+1, vi〉vkvi. (37)
The predictor of Xn+1 is now given by Eq. (28), Eq. (35) and Eq. (37) as
Xn+1 = Ψp(Xn) =
∫ 1
0ψp(t, s)Xn(s)ds =
=
p∑k=1
(p∑l=1
ψkl〈Xn, vl〉
)vk(t),
(38)
where
ψji = λ−1i1
N − 1
N−1∑n=1
〈Xn, vi〉〈Xn+1, vj〉. (39)
18
4.4 Model formulation
In order to evaluate whether FDA techniques can be used for risk management
purposes, we propose two different models and compare them with empirical yield
data. The first model uses the finite approximation of the Karhuen-Loeve expansion
given in Eq. (20) and has the form
Xn(t) =
p∑k=1
σkZnk vk(t) (40)
where σk for k = 1, 2, ..., p are the standard deviations of the principal component
scores ξnk = 〈Xn(t), vk(t)〉 and Znk are standard normally distributed variables such
that Cov(Znk , Znl ) = 0 for k 6= l and Cov(Znk , Z
mk ) = 0 for m 6= n. The model
implicitly assumes that the principal component scores ξnk are normally distributed
with mean zero and standard deviation σk and that yield curve shifts are independent
across time.
The second model uses the theory underlying FAR(1) operators as described in
Section 4.3 and has the form
Xn(t) = Ψp(Xn−1) +
p∑k=1
σ∗kZnk v∗k(t) (41)
where Ψp is the empirical FAR(1) operator given in Eq. (35). The second term
in this expression represents the error term εn(t) in Eq. (28). In particular, v∗k(t)
denotes the principal components of the error term and σ∗k denotes the standard
deviations of the corresponding scores ξn∗k = 〈εn(t), v∗k(t)〉. In contrast to the model
described by Eq. (40), this model does not assume independent yield curve shifts
across time as implied by the FAR(1) operator. The model however assumes that
the errors are temporally independent.
19
4.5 Methods for evaluating model performance
Let X∗j (ti) denote ordered observations of yield changes X(ti) such that X∗1 (ti) <
X∗2 (ti) < ... < X∗N (ti). Furthermore, let Y ∗j (ti) denote corresponding simulated
(model) yield curve shifts such that Y ∗1 (ti) < Y ∗2 (ti) < ... < Y ∗N (ti). It is assumed
that X∗j (ti) and Y ∗j (ti) are outcomes of the stochastic variables X∗(ti) and Y ∗(ti)
with probability distribution functions FX∗(ti) and FY ∗(ti) respectively. In order
to evaluate whether the artificial distributions FY ∗(t1), FY ∗(t2), ..., FY ∗(tm), appear
similar to the true empirical distributions FX∗(t1), FX∗(t2), ..., FX∗(tm), we use two
different evaluation techniques.
Firstly, we visually inspect the marginal distributions of yield changes for each ma-
turity by illustrating them in quantile-quantile plots (henceforth referred to as qq-
plots). This is done by plotting the points {F−1X∗(ti)
(N−k+1N+1
), F−1Y ∗(ti)
(N−k+1N+1 ) : k =
1, 2, ..., N} in a diagram. If FY ∗(ti) = FX∗(ti), the points in the qq-plots should form
a straight line in the diagram.
Secondly, we complement the qq-plot inspection by conducting the Kolmogorov-
Smirnov [10] test of the null hypothesis H0 : FX∗(ti) = FY ∗(ti) against the alternative
H1 : FX∗(ti) 6= FY ∗(ti). This is done by considering the test statistic
D =
√N
2+ 0.12 +
0.11√N2
d (42)
where d = max(FX∗(ti)(s)− FY ∗(ti)(s)
)is the maximum difference between the
empirical distribution functions
FX∗(ti)(s) =1
N
N∑j=1
I{X∗j (ti)<s}(s) (43)
and
FY ∗(ti)(s) =1
N
N∑j=1
I{Y ∗j (ti)<s}(s) (44)
20
of X∗ and Y ∗ respectively. Here I{A}(s) =
1 if s ∈ A
0 if s 6∈ Adenotes the indicator
function. Under H0, the probability to observe the value d or larger is given by
p(D) = 2∑∞
k=1(−1)k−1e−2k2D2
. Hence, H0 may be rejected if this probability is
small.
The qq-plot and the Kolmogorov-Smirnov test allow us to evaluate whether any
of the two simulation models capture the marginal distribution of observed yields.
However, a successful simulation model should also capture the joint behavior of
yields with different maturities. In order to analyze this property, we compare
cross-sectional correlation matrices between empirical and simulated samples.
5 Results
5.1 Empirical functional principal components
The first 10 principal component functions vk(t), k = 1, ..., 10 are illustrated in
Figure 4, whereas Table 3 reports estimated eigenvalues λk as well as cumulative
fractions of total variance explained up to each principal component. Results suggest
that 86.9% of the observed variance is explained by the first principal component,
implying that it is possible to capture much of the observed yield curve dynamics
with only one stochastic factor. However, if 10 principal components are used, we
may explain as much as 99.9% of the observed variance as shown in Table 3.
Histograms of the corresponding principal component scores ξnk = 〈Xn, vk(t)〉 are
illustrated in Figure 5 which suggests that the distributions of ξnk appear symmetrical
and centered around zero.
In order to analyze the severity of the estimation error of the finite Karhuen-
Loeve approximation in Eq. (20), we plot the estimation errors en(t) = Xn(t) −∑pk=1 ξ
nk vk(t) for p = {2, 6, 10} in Figure 6. As expected, the basis expansion ap-
pears to better approximate yield curve shifts for large values of p implied by the
21
00.
10.
20.
30.
40.
50.
60.
70.
80.
91
−10−
5051015
t
vk(t)
k=1
k=2
k=3
k=4
k=5
k=6
k=7
k=8
k=9
k=10
Fig
ure
4:
Funct
ional
pri
nci
pal
com
ponen
ts,
equiv
ale
nt
toth
efirs
t10
eigen
funct
ions
of
the
cova
riance
op
erato
r.
22
−0.
50
0.5
05101520k=
1
−0.
050
0.05
01020304050k=
6
−0.
50
0.5
0510152025k=
2
−0.
050
0.05
05101520253035k=
7
−0.
50
0.5
0510152025303540k=
3
−0.
050
0.05
05101520253035k=
8
−0.
50
0.5
01020304050k=
4
−0.
050
0.05
01020304050k=
9
−0.
50
0.5
0510152025303540k=
5
−0.
050
0.05
01020304050k=
10
Fig
ure
5:F
unct
ional
pri
nci
pal
com
ponen
tsc
ore
sco
rres
pondin
gto
the
firs
t10
eigen
funct
ions
of
the
cova
riance
op
erato
r.N
ote
the
diff
eren
tsc
aling
of
thex
axes
bet
wee
nth
eupp
erpanel
and
the
low
erpanel
.
23
k λk Fraction of total variance1 0.009384363 86.9%2 0.000838669 94.7%3 0.000234191 96.9%4 0.000160627 98.4%5 0.000072049 99.0%6 0.000039878 99.4%7 0.000016059 99.5%8 0.000013559 99.7%9 0.000013211 99.8%10 0.000010272 99.9%
Table 3: The first 10 eigenvalues λk and the total cumulative variance explained up to each k.
smaller errors. In addition, it seems that the estimation error is higher for shorter
maturities compared to the medium-term and long-term counterparts, regardless of
the choice of p.
To gain further understanding of the error characteristics we plot means, standard
deviations, maximum values and minimum values of e(t) for different maturities and
different choices of p in Figure 7. Results suggest that the mean error is virtually
zero for all choices of p and for all maturities, implying that the method does not
introduce any systematic estimation bias. Furthermore, error standard deviations
and maximum errors (in absolute terms) decrease with p which confirms that the
approximation improves with p. Figure 7 however, again, suggests that the approxi-
mation error is more severe for short-term maturities implied by the larger standard
deviations and maximum errors for the 3 month rate.
5.2 FAR(1) predictions
Given the results above, the empirical FAR(1) operator Ψp can be calculated using
Eq. (38). Predicted functions Xn and the corresponding prediction errors εn(t)
are plotted in Figure 8 using p = 10 eigenfunctions and the full data set of 590
observations. Visual inspection of Figure 8 suggests that Ψp captures much of the
observed curvature in yield curve shifts around short maturities. In particular, the
resulting errors εn(t) appear more flat compared to the original data shown in Figure
3. Furthermore, results imply that some of the outliers observed in the original data
24
00.
10.
20.
30.
40.
50.
60.
70.
80.
91
−1012
t
e(t)
p=2
00.
10.
20.
30.
40.
50.
60.
70.
80.
91
−0.
50
0.51
t
e(t)
p=6
00.
10.
20.
30.
40.
50.
60.
70.
80.
91
−0.
2
−0.
10
0.1
0.2
t
e(t)
p=10
Fig
ure
6:E
stim
ati
on
erro
rari
sing
from
repla
cing
the
infinit
eK
arh
uen
-Loev
eex
pansi
on
wit
hit
sfinit
eco
unte
rpart
forp
=2,6,1
0.
Note
the
diff
eren
tsc
ale
son
they-a
xes
.
25
23
45
67
89
10−
10−8
−6
−4
−2024
x 10
−18
p
Mean error
23
45
67
89
100
0.02
0.04
0.06
0.080.
1
p
Error standard deviation
03
M01
Y05
Y10
Y30
Y
23
45
67
89
100
0.2
0.4
0.6
0.81
1.2
1.4
p
Maximum errors
23
45
67
89
10−
0.5
−0.
4
−0.
3
−0.
2
−0.
10
p
Minimum errorsF
igu
re7:
Mea
n,
standard
dev
iati
on,
maxim
um
valu
esand
min
imum
valu
esfo
rth
eappro
xim
ati
on
erro
rof
sele
cted
yie
lds
ari
sing
from
repla
cing
the
Karh
uen
-Loev
eex
pansi
on
by
its
finit
eco
unte
rpart
.R
esult
sare
plo
tted
for
diff
eren
tva
lues
ofp.
26
set are captured by Ψp since the dispersion of the errors seems to be somewhat
smaller than the original yield curve shifts.
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
t
Ψp(X
n(t))
FAR(1) predictions
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
t
ε n(t)
FAR(1) errors
Figure 8: The upper plot shows FAR(1) predictions Xn(t) = Ψp(Xn−1(t)), whereas the lower plot
shows the corresponding estimation errors εn(t) = Ψp(Xn−1(t))− Xn(t).
5.3 Simulation model performance
In this section, we evaluate the performance of the models given in Eq. (40) and Eq.
(41). Results from simulating 590 yield curve shifts for both models are illustrated
in Figure 9. The σk and σ∗k coefficients in these simulations were set to their corre-
sponding sample estimates. To facilitate comparison, Figure 9 also shows empirical
yield curve observations already shown in Figure 3.
In order to evaluate the in- and out-of-sample performance of each simulation model,
we split the sample into one estimation sample containing the first 295 observations
of the data set and one validation sample containing the remaining 295 observations.
We thereafter extract the first 10 principal components of the estimation sample and
estimate the corresponding parameters σk and σ∗k. Finally, we simulate 295 yield
27
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5Simulated independent yield curve shifts
t
Xn(t
)
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
t
Xn(t
)
Simulated FAR(1) yield curve shifts
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5Real yield curve shifts
t
Xn(t
)
Figure 9: Comparison between simulated and observed shifts. The upper plot shows curvessimulated according to Eq. (40) which assumes that shifts are independent across time. Themiddle plot shows curves simulated according to Eq. (41) which assumes that yield curve shiftsobey the FAR(1) process with independent errors. The lower plot shows real yield curve shifts.
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
t
p−va
lue
Kolmogorov−Smirnov test (in−sample)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
t
p−va
lue
Kolmogorov−Smirnov test (out−of−sample)
Independent modelFAR(1) model
Figure 10: Significance levels of the Kolmogorov-Smirnov test.
28
curve shifts of each model and compare these with the estimation sample (in-sample
comparison) and the validation sample (out-of-sample comparison).
Figure 11 shows in-sample qq-plots for a selection of maturities for both simulation
models. Visual inspection of these plots suggests that distributions of simulated
yield shifts appear similar to the empirical distributions for medium-term and long-
term maturities. In particular, the FAR(1) model performs slightly better than the
independent yield curve shift for the 5-year, 10-year and 30-year maturities. None
of the models, however, perform satisfactory for the two shortest maturities.
Figure 12 shows the corresponding out-of-sample qq-plots. While both models ap-
pear to more or less replicate the distributions of 10-year yields and 30-year yields,
they perform somewhat worse for 5-year yields compared to their in-sample per-
formances. In addition, the poor performance for short maturities remains in the
out-of-sample comparison.
Significance levels obtained from Kolmogorov-Smirnov tests are plotted for t =
0, 0.01, ..., 1 in Figure 10 which confirms the poor distributional fits between simu-
lated and empirical data at the short end of the yield curve. The results show small
p values for short-term maturities both in- and out-of-sample, suggesting that the
null hypothesis of equal distributions can be rejected at this segment of the yield
curve.
Finally, Table 4 displays the differences in estimated sample cross-sectional correla-
tions between the empirical sample and each of the two simulation samples. Small
values of these differences suggest that the simulation samples appear to replicate
the cross-sectional dependence of yield changes. Results show that this error ranges
in the interval [−0.047, 0.121] for the independent yield curve model, whereas the
corresponding interval for the FAR(1) model is [−0.009, 0.0078]. Hence, the FAR(1)
model appears slightly better at capturing cross-sectional correlations, although both
models can be said to perform satisfactory in this respect.
29
−2
−1
01
−6
−5
−4
−3
−2
−1012
Independent model
03M
−2
−1
01
−5
−4
−3
−2
−101
FAR(1) model
03M
−1
−0.
50
0.5
−1.
5
−1
−0.
50
0.51
01Y
−1
−0.
50
0.5
−1.
5
−1
−0.
50
0.51
01Y
−0.
50
0.5
−0.
50
0.5
05Y
−0.
50
0.5
−0.
50
0.5
05Y
−0.
50
0.5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
10Y
−0.
50
0.5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
10Y
−0.
50
0.5
−0.
50
0.5
30Y
−0.
50
0.5
−0.
6
−0.
4
−0.
20
0.2
0.4
30Y
Fig
ure
11:
In-s
am
ple
qq-p
lots
bet
wee
nth
edis
trib
uti
ons
of
sim
ula
ted
yie
ldch
anges
and
true
yie
ldch
anges
of
sele
cted
matu
riti
es.
The
upp
erpanel
corr
esp
onds
tosi
mula
ted
yie
ldch
anges
acc
ord
ing
toE
q.
(40)
wher
eas
the
low
erpanel
corr
esp
onds
tosi
mula
tions
of
Eq.
(41).
30
−0.
50
0.5
−1.
5
−1
−0.
50
0.51
Independent model
03M
−0.
50
0.5
−1.
5
−1
−0.
50
0.51
FAR(1) model
03M
−0.
50
0.5
−1.
5
−1
−0.
50
0.51
01Y
−0.
50
0.5
−1.
5
−1
−0.
50
0.51
01Y
−0.
50
0.5
−0.
8
−0.
6
−0.
4
−0.
20
0.2
0.4
0.6
05Y
−0.
50
0.5
−0.
8
−0.
6
−0.
4
−0.
20
0.2
0.4
0.6
05Y
−0.
50
0.5
−0.
4
−0.
20
0.2
0.4
0.6
10Y
−0.
50
0.5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
10Y
−0.
50
0.5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
30Y
−0.
50
0.5
−0.
4
−0.
3
−0.
2
−0.
10
0.1
0.2
0.3
0.4
30Y
Fig
ure
12:
Out-
of-
sam
ple
qq-p
lots
bet
wee
nth
edis
trib
uti
ons
of
sim
ula
ted
yie
ldch
anges
and
true
yie
ldch
anges
of
sele
cted
matu
riti
es.
The
upp
erpanel
corr
esp
onds
tosi
mula
ted
yie
ldch
anges
acc
ord
ing
toE
q.
(40)
wher
eas
the
low
erpanel
corr
esp
onds
tosi
mula
tions
of
Eq.
(41).
31
3M6M
1Y2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y
10Y
20Y
30Y
Sim
ula
ted
ind
epen
den
tyie
ldcu
rve
shif
ts3M
0.00
00.
057
0.07
50.
121
0.0
87
0.0
85
0.0
86
0.0
74
0.0
63
0.0
49
0.0
35
0.0
28
0.0
06
-0.0
14
6M0.
057
0.00
00.
009
0.05
10.0
26
0.0
23
0.0
24
0.0
40
0.0
31
0.0
20
0.0
06
0.0
04
-0.0
07
-0.0
17
1Y0.
075
0.00
90.
000
0.01
90.0
07
0.0
00
-0.0
07
0.0
01
-0.0
07
-0.0
17
-0.0
30
-0.0
31
-0.0
39
-0.0
47
2Y0.
121
0.05
10.
019
0.00
00.0
03
-0.0
04
-0.0
06
-0.0
05
-0.0
09
-0.0
12
-0.0
17
-0.0
21
-0.0
27
-0.0
27
3Y0.
087
0.02
60.
007
0.00
30.0
00
0.0
07
0.0
01
0.0
02
0.0
01
0.0
01
-0.0
01
-0.0
03
-0.0
01
-0.0
01
4Y0.
085
0.02
30.
000
-0.0
040.0
07
0.0
00
0.0
08
0.0
03
0.0
02
0.0
02
0.0
00
-0.0
01
0.0
00
0.0
03
5Y0.
086
0.02
4-0
.007
-0.0
060.0
01
0.0
08
0.0
00
0.0
01
-0.0
01
-0.0
02
-0.0
05
-0.0
05
-0.0
07
-0.0
01
6Y0.
074
0.04
00.
001
-0.0
050.0
02
0.0
03
0.0
01
0.0
00
0.0
01
0.0
02
0.0
00
-0.0
01
-0.0
01
-0.0
04
7Y0.
063
0.03
1-0
.007
-0.0
090.0
01
0.0
02
-0.0
01
0.0
01
0.0
00
0.0
02
0.0
01
0.0
00
0.0
00
-0.0
02
8Y0.
049
0.02
0-0
.017
-0.0
120.0
01
0.0
02
-0.0
02
0.0
02
0.0
02
0.0
00
0.0
02
0.0
02
0.0
00
0.0
01
9Y0.
035
0.00
6-0
.030
-0.0
17-0
.001
0.0
00
-0.0
05
0.0
00
0.0
01
0.0
02
0.0
00
0.0
00
-0.0
03
0.0
03
10Y
0.02
80.
004
-0.0
31-0
.021
-0.0
03
-0.0
01
-0.0
05
-0.0
01
0.0
00
0.0
02
0.0
00
0.0
00
0.0
01
0.0
07
20Y
0.00
6-0
.007
-0.0
39-0
.027
-0.0
01
0.0
00
-0.0
07
-0.0
01
0.0
00
0.0
00
-0.0
03
0.0
01
0.0
00
0.0
07
30Y
-0.0
14-0
.017
-0.0
47-0
.027
-0.0
01
0.0
03
-0.0
01
-0.0
04
-0.0
02
0.0
01
0.0
03
0.0
07
0.0
07
0.0
00
Sim
ula
ted
FA
R(1
)yie
ldcu
rve
shif
ts3M
0.00
00.
009
0.00
90.
025
0.0
05
0.0
18
0.0
23
0.0
18
0.0
23
0.0
32
0.0
42
0.0
38
0.0
61
0.0
53
6M0.
009
0.00
00.
004
0.02
00.0
31
0.0
38
0.0
36
0.0
43
0.0
46
0.0
48
0.0
54
0.0
44
0.0
68
0.0
74
1Y0.
009
0.00
40.
000
0.01
70.0
43
0.0
55
0.0
51
0.0
51
0.0
56
0.0
61
0.0
69
0.0
68
0.0
77
0.0
78
2Y0.
025
0.02
00.
017
0.00
00.0
13
0.0
08
0.0
06
0.0
05
0.0
07
0.0
09
0.0
10
0.0
10
0.0
03
0.0
05
3Y0.
005
0.03
10.
043
0.01
30.0
00
0.0
06
-0.0
01
-0.0
03
-0.0
02
-0.0
03
-0.0
03
-0.0
03
-0.0
09
-0.0
09
4Y0.
018
0.03
80.
055
0.00
80.0
06
0.0
00
0.0
07
0.0
00
0.0
00
0.0
02
0.0
02
0.0
02
-0.0
04
-0.0
02
5Y0.
023
0.03
60.
051
0.00
6-0
.001
0.0
07
0.0
00
0.0
01
0.0
01
0.0
03
0.0
03
0.0
03
-0.0
02
0.0
00
6Y0.
018
0.04
30.
051
0.00
5-0
.003
0.0
00
0.0
01
0.0
00
0.0
02
0.0
03
0.0
02
0.0
02
0.0
00
0.0
00
7Y0.
023
0.04
60.
056
0.00
7-0
.002
0.0
00
0.0
01
0.0
02
0.0
00
0.0
02
0.0
00
0.0
01
-0.0
02
-0.0
01
8Y0.
032
0.04
80.
061
0.00
9-0
.003
0.0
02
0.0
03
0.0
03
0.0
02
0.0
00
0.0
01
0.0
02
-0.0
02
-0.0
01
9Y0.
042
0.05
40.
069
0.01
0-0
.003
0.0
02
0.0
03
0.0
02
0.0
00
0.0
01
0.0
00
0.0
00
-0.0
04
-0.0
01
10Y
0.03
80.
044
0.06
80.
010
-0.0
03
0.0
02
0.0
03
0.0
02
0.0
01
0.0
02
0.0
00
0.0
00
-0.0
05
-0.0
04
20Y
0.06
10.
068
0.07
70.
003
-0.0
09
-0.0
04
-0.0
02
0.0
00
-0.0
02
-0.0
02
-0.0
04
-0.0
05
0.0
00
0.0
09
30Y
0.05
30.
074
0.07
80.
005
-0.0
09
-0.0
02
0.0
00
0.0
00
-0.0
01
-0.0
01
-0.0
01
-0.0
04
0.0
09
0.0
00
Tab
le4:
Diff
eren
ceb
etw
een
esti
mate
dco
rrel
ati
on
coeffi
cien
tsfo
rth
etr
ue
sam
ple
and
the
two
sim
ula
ted
sam
ple
s.
32
6 Conclusion
This thesis provides a framework which can be used for yield curve risk manage-
ment purposes. By modeling yield curve shifts as distinct but continuous stochastic
objects defined over a continuum of maturities, we describe how the theory of FDA,
FPCA and FAR operators can be applied in this setting. Based on these tech-
niques, we provide two stochastic models which attempt to describe realistic yield
curve shifts based on observed historical data.
The first of these models assumes that yield curve shifts are independent across time,
whereas the second model incorporates temporal dependence through the introduc-
tion of the FAR(1) operator. We find that both models more or less replicate the
marginal distributions of yield changes for medium-term and long-term maturities,
although the FAR(1) model appears to perform slightly better for this segment of the
yield curve. None of the models however performs satisfactory at the short-maturity
segment of the yield curve.
This failure may be generated by the approximation error arising from replacing the
infinite Karhuen-Loeve expansion by its finite counterpart. Indeed, this is supported
by the in-sample assessment of the approximation error (described in Section 5.1)
which suggests that the short end of the yield curve is more sensitive to small values
of p. Non-normality of short-maturity yield changes may be another likely reason
for the failure to capture these dynamics. It remains a task for future research to
determine the exact marginal distribution of principal component scores and yield
curve shifts.
Both of the proposed models appear to capture much of the cross-sectional depen-
dence observed across the yield curve as measured by sample correlations. This
measure likely provides a general idea of how yields co-move across the curve, al-
though it only provides information about linear dependence. Hence, future research
may improve the analysis by considering more general measures of dependence.
33
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34
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