Czech Yield Curve
WORKING PAPER No. 11/2010
Martin Cícha
December 2010
Řada studií Working Papers Centra výzkumu konkurenční schopnosti
české ekonomiky je vydávána s podporou projektu MŠMT výzkumná
centra 1M0524.
ISSN 1801-4496
Vedoucí: prof. Ing. Antonín Slaný, CSc., Lipová 41a, 602 00 Brno,
e-mail: slany@econ.muni.cz, tel.: +420 549491111
Abstract:
We present a yield curve model based on principal component
analysis and nonlinear stochastic differential equations. The model is
applied everywhere where quantification of interest rate dynamics is
needed, i.e. in the Monte Carlo risk management of interest rate
instruments or in the evaluation of profitability and risk of interest rate
derivatives and different business strategies. Applying presented
model, we are also able to price the interest rate derivatives or predict
the distribution of interest costs.
Keywords:
interest rate models, yield curve, principal component analysis,
nonlinear stochastic differential equations, risk management
Recenzoval:
Ing. Irena Jindřichovská, CSc.
4
1. INTRODUCTION
The yield curve modeling is very important area of financial risk
management. Especially during the last few years, we could observe
high volatility of interest rates. Lot of hedge funds, pension funds,
investment banks, debt offices and other assets and debt managers
were affected by this increasing volatility. The yield of corporate and
government bonds have increased significantly during the financial
crisis. Due to current debt crisis on the periphery of European
monetary union, bond yields remain at high level. These high bond
yields are in contrast with historically low money market rates and
caused by increasing credit spread of issuers. Thus credit spread of
issuer can be expressed by the difference between bond yield and
appropriate money market rate. If we want to understand and to
manage all interest cost we need to understand the dynamic of money
market rates first. Money market rates are also used as reference rate
which is further reason for understanding their dynamics.
In this paper, we present a money market yield curve model based on
principal component analysis and nonlinear stochastic differential
equations. Thus we describe the dynamics of money market rates
which can be applied everywhere where the interest rate risk needs to
be manage. Applying presented model we are able to price the interest
rate derivatives or predict the distribution of interest costs.
We design the presented model for future simulation of yield curves,
which can be used, for example, for prediction of yield curves. This
paper is based on [1] but updated and more suitable for interest costs
estimate. In the model, we add some more information in terms adding
daily data. We also include high volatile data from period of financial
crisis. Further, we decrease the time step for better description of
interest rates dynamics.
1.1. Previous research in the area of modeling
the yield curve
In the past we have tried to describe the money market yield curve
through short rate models. These models are based on unobservable
short rate. Among the famous models of short rate belong Vasicek,
CIR (Cox-Ingersoll-Ross), (CKLS Chan-Karolyi-Longstaff-Sanders) etc.
The short rate models are primarily designed for pricing of interest rate
derivatives. The short rate models describe only the dynamics of short
rate and the dynamics of the other observable rates needs to be
determined from the short rate dynamics. We have found that these
models are not suitable for interest cost estimates since they
underestimate the volatility of longer tail of yield curve significantly.
The presented model describes the dynamics of the whole yield curve
at once which is desirable and removes the volatility underestimation
problem.
5
2. STOCHASTIC DIFFERENTIAL
EQUATIONS AND THEIR APPLICATION
IN YIELD CURVE MODELING
The application of stochastic processes in continuous time in financial
modeling has become popular. The financial model is usually based on
the diffusion process
{ }0, ≥tXt . The dynamics of this process is
determined by following stochastic differential equation (SDE):
,)()(= tttt dWXdtXdX σµ +
(1)
where
{ }0, ≥tWt is the standard Brownian motion. The functions
)(⋅µ
and
)(⋅σ are called drift and diffusion of the process, respectively. We
assume the following parameterization of SDE:
),,(=)(),(=)( 22
θσσθµµ xxaxx
(2)
where θ is the parameter vector of parametric space
K
R⊂Θ . In this
paper we consider only Markov (i) and strictly stationary processes (ii).
The Markov process is defined as a process, for which in time t the
conditional distribution function of future values sX
, ts > depends only
on tX
(i.e. is independent of the historical values
tuXu <,
.). A
process is strictly stationary when joint distribution holds for all m , for
all time sequences
∞<<<<0 1 mtt L
and for all mxx ,,1 K
from state
space following equation
),,( 11 mmtt xXxXP ≤≤ K
=
),,( 11 mmtt xXxXP ≤≤
∆+∆+
K
. Drift and
diffusion functions fully specify the model. A frequent choice of the drift
for interest rate models is a linear function
)(=),( xx −µαθµ , where the
rate x returns to its mean value
µ with the speed α . This property is
called mean reversion and it is the fundamental postulate of interest
rate models. Diffusion function differentiates then among the models.
For instance a Vasicek model uses constant diffusion
22
=),( σθσ x
the
diffusion function of the CIR model is
xx 22
=),( σθσ .
The marginal (unconditional) density of a stationary process is 1
,
),(
),(2
exp
),(
)(
=),( 2
0
2






∫ du
u
u
x
x
x
x θσ
θµ
θσ
θξ
θπ
(3)
1
Marginal density equation comes from Fokker-Planck equation, see [4] or more
precisely see [2].
6
where
)(θξ is the normalization constant. The lower bound 0x
of the
integral is not unique. The choice of 0x
is neutralized by the
normalization constant, which warrants that the integral through the
domain of definition of the process is equal to one.
Using the SDE for modeling financial processes, we first need to
specify drift and diffusion functions of the SDE and to define the
parametric space Θ . Secondly, we need to estimate the parameters of
the specified model.
The definition of the parametric space Θ comes from the domain of
definition of the process
),(= xxD
, from the postulates (i) and (ii) and
from specific claims on behavior of the process (e.g. mean reversion).
In this paper we will consider the process of principal components of a
yield curve. A process of principal component is defined for all real
numbers:
).,(= ∞−∞D
The definition of the Principal Component
Analysis implies that the expected value of each principal component
is equal to zero. We demand mean reversion property of the given
process, which is guaranteed if
0.>),(lim0<),(lim θµθµ xandx
xx −∞→∞→
(4)
Following cubic drift and constant diffusion are used in our model:
,=)(=)( 0
23
3
2
210 βσααααµ xandxxxx +++
(5)
where
),,,,(= 03210 βααααθ .
The solution (4) for cubic drift (5) can be found out that mean reversion
is guaranteed, when
0<3α .
Our choice of cubic drift for principal component of yield curve
modeling support the time series of principal components of Czech and
European yield curves. The choice of constant diffusion function
makes practical application of the model easier due to analytical
solution of the integral in the exponent of equation (3).
3. ESTIMATION OF PARAMETERS OF
SDE
We based the parameter estimation of SDE on minimizing the distance
between the parametric density of the SDE and its nonparametric
estimation (empirical density). Kernel density (see [5]) is used for
nonparametric density estimation. We estimate the nonparametric
density from time series
},,{ 1 nxx K
, where observations are recorded in
7
time interval ∆ (i.g. ∆ = 1 month). We search for parameter vectorθ ,
which minimize the distance between parametric density of the SDE
given by the equation (3) and the nonparametric density estimation.
This idea was first introduced by Yacine Ait-Sahalia in [4]. Yacine AitSahalia
estimated the so-called General Parametric Model of three
months Eurodollar interest rate. General Parametric Model defines the
dynamics of interest rate by the following SDE:
ttttttt dWrrdtrrrdr )()/(= 3
2103
2
210
β
βββαααα ++++++
.
As we mentioned above, we estimate the parameters of SDE from
observations without discretizing the SDE. The term discretisation
refers to the approximation of a continuous process with a discrete
process. It is clear that discretisation always brings bias, which
increases with the length of the observation interval ∆ . If we have high
frequency time series, where 0→∆ , the bias caused by discretisation
is insignificant. For financial time series recorded on daily or even
monthly basis, the discretisation can be a too inaccurate approximation
of the original, time-continuous process. Under these conditions the
bias could be no longer tolerable.
3.1. Nonparametric estimation of marginal
density
The nonparametric estimation of marginal density is given by:
,
11
=)(ˆ
1=







 −
∑ n
i
n
n
i
b
xu
K
bn
uπ
(6)
where
)(⋅K
is the kernel function and nb
is the smoothing constant
(bandwidth). The equation (6) can be interpreted as a "continuous
histogram". The choice of nb
is crucially important for the smoothing
level of the density - higher nb
causes higher level of smoothing.
Setting the optimal value of nb
is difficult for real market data. Some
methods for choosing optimal nb
were introduced but these assume
independent, identically distributed (i.i.d.) data. The financial data are
usually strongly auto correlated. Due to the abovementioned problems,
we determine the value of nb
by following empirical rule:
,= 5
1
−
ncbn σ (7)
where σ is the standard deviation of time series, n is the number of
observations and c is a constant. For independent normal distributed
8
data, the recommended value of c is 1,05. We use the normal kernel
function
π2/2)/(=)( 2
uexpuK −
.
3.2. Marginal density of SDE
The easiest practically applicable SDE is the linear drift with constant
diffusion (Vasicek model). Marginal density of the corresponding SDE
is normally distributed. The marginal distribution for the combination of
linear drift and diffusion directly proportional to the value of the process
(CIR model) is gamma distribution. The cubic drift and constant
diffusion given in (5) has marginal density defined in (3). The marginal
density can be adjusted to the form:
,
23
22
exp
)(
=),(
0
4
3
0
3
2
0
2
1
0
0
0 







+++
β
α
β
α
β
α
β
α
β
θξ
θπ
xxx
x
(8)
where
)(θξ is chosen so that the integral of marginal density (8) over
the process domain of definition
),(= xxD
is equal to one (in the case
of principal component
),(= +∞−∞D
).
3.3. Objective function of estimation
Estimation is based on the minimization of the distance between the
marginal density of SDE (parametric density) given in equation (3) and
its nonparametric estimation (empirical density) given in equation (6).
We define the objective function of the estimation the same way as in
[4] in terms of mean square error (MSE) defined by:








− 2
))(ˆ),((= XXEO πθπ
.)(ˆ))(ˆ),((= 2
duuuu
x
x
ππθπ −∫ (9)
The parameters θ are estimated on the basis of observations
},,{ 1 nxx K
minimizing the sample objective function:
.))(ˆ),((
1
minarg=ˆ 2
1=
ii
n
i
xx
n
πθπθ
θ
−∑Θ∈
(10)
The estimate θˆ is asymptotically normal:
9
),(0,)ˆ( Ω→− Nn
d
θθ (11)
where Ω and its estimation are described on page 420 in [4].
Practically it is profitable to handle the logarithm of marginal densities,
which is easier to optimize. Using the cubic drift and constant diffusion
given in (8) we solve
.)(ˆlog
23
22
log)(log
1
minarg=ˆ
2
0
4
3
0
3
2
0
2
1
0
0
0
1=








−++++−∑Θ∈
i
iiii
n
i
x
xxxx
n
π
β
α
β
α
β
α
β
α
βθξθ
θ
(12)
The domain of integration of constant
)(θξ can be determined by
extending sufficiently the range of the observations
},,{ 1 nxx K
. We
have identified that adding 1.5times the length of the interval to the
maximum and subtracting it from the minimum is mostly sufficient.
4. PRACTICAL APPLICATION
We apply the cubic drift and constant diffusion SDE on the Czech yield
curve (YC). We don't model directly the YC process, but we apply the
principal component analysis. We also look at the process behavior
with a simulation experiment.
4.1. The data
The estimation of parameters is performed on the basis of daily
observations. We used maturities of 3M, 6M, 9M, 12M, 3Y, 5Y, 10Y
and 15Y Czech zero-coupon yield curve from August 29, 2000 until
October 18, 2010. We drop the nontrading days, i.e. for each maturity
we have 2554 observations. Maturities shorter than one year are rates
of PRIBOR, maturities over one year are rates of swap market. All data
are downloaded from Bloomberg. The yield curve is displayed in figure
1.
Figure 1: Czech yield curve since 29.08.2000 till 18.10.2010.
Maturities are displayed in months, values 3, 6, 9 a 12 present 3M
PRIBOR - 12M PRIBOR, values 36, 60 a 120 present 3Y, 5Y 10Y and
15Y swap. Y - axis presents interest rate in percents.
10
The variability of the series tends to be dependent on its level, so we
performed logarithmic transformation on the series. The relationship
between variability and level of interest rates and effectiveness of the
logarithmic transformation are shown in figure 2. It is obvious that the
logarithmic transformation decreases dependence of the variability of
the rates on their level and stabilizes them in term of variance.
Figure 2: The relationship between the variability and the level of
interest rates. The time series for all maturities are divided into 9 day
periods. For each period we calculate the mean (horizontal axis) and
the standard deviation (vertical axis). The upper chart: no
transformation; the bottom chart: logarithmic transformation. The line
represents the linear trend.
11
4.2. Principal component analysis
Direct yield curve modeling would represent multivariate stochastic
process modeling of highly correlated variables. We solve this problem
by applying the principal component analysis. This method is a linear
transformation of input variables to new variables - so called
components. Input variables are linear combinations of components.
The weights of the combination are called component loadings. The
aim of the transformation is to reduce the size of the data, while
preserving as much information contained in the data as possible. The
first three components of the Czech YC logarithms explain over 99.7 %
of its variability. The first component explains 93.6 % of the variability
of the YC. Another advantage of the principal components analysis is
that the components are (following from its definition) independent.
That means that we can model the components separately with
univariate processes. We denote the first three components with 1PC
,
2PC
and 3PC
.
The upper chart of figure 3 displays the component loadings for
different maturities. The component loadings of the first principal
component are approximately horizontal. We can interpret the first
components as the level factor of the interest rates. Changes in the
first component cause a parallel shift of the YC. Component loadings
of the second component are increasing and changes in the second
component cause rotation of the YC. We can interpret the second
component as a slope factor. The component loadings of the third
component are "hump" shaped and changes in this component cause
change of the shape of the YC. We can interpret this component as a
curvature factor. This interpretation of components was first presented
in [3].
Figure 3: Component loadings and time series of the first three
components of the Czech YC.
Top: The component loadings - maturities in months are on the
horizontal axis.
Bottom: Time series of the components from August 29, 2000, until
October 18, 2010.
12
The bottom chart of figure 3 displays the time series of the components
from August 29, 2000, until October 18, 2010. The components are
calculated from the logarithms of the Czech YC (figure 1). It follows
from the definition of the principal components that its expected value
is zero.
13
4.3. The estimate of cubic drift of the principal
components
To quantify the dynamics of the first three principal components 1PC
,
2PC
and 3PC
, we estimate parameters of the SDE with cubic drift and
constant diffusion (formula (5)). The first step is the nonparametric
estimate of the probability distribution function using the kernel density
defined in (6). Then we estimate the parameters of the SDE,
minimizing the difference between parametric marginal distribution (8)
(corresponding to the SDE with cubic drift and constant diffusion) and
the nonparametric estimate of the marginal distribution. The objective
function is defined in the formula (12). We take the following constrains
into consideration:
0<3α and
0>0β . The parametric and
nonparametric estimates of the marginal distribution function of the first
component are displayed in bottom chart of figure 4. The upper left
chart displays the cubic drift and the observed daily differences of the
time series of 1PC
. The upper right chart displays the volatility and the
observed volatility of the time series of 1PC
. The same figures
concerning 2PC
and 3PC
are in displayed in figures 5 and 6,
respectively.
Figure 4:
Upper left chart: The cubic drift and the observed daily differences of
PC1 (dots).
Upper right chart: The constant volatility and the observed volatility of
PC1 (dots).
Bottom chart: The estimated optimal parametric and the nonparametric
distribution function of PC1 of the Czech YC. Horizontal axis - the
values of PC1, vertical axis - values of the marginal probability
distribution function.
14
15
Figure 5: The same situation as in figure 4 but for the component PC2
of the Czech YC.
16
Figure 6: The same situation as in figure 4 but for the component PC3
of the Czech YC
The parameter estimates, value of constant c needed in formula (7)
are displayed in table 1. The value of constant c was set arbitrarily.
The nonparametric density estimate has to be smooth enough, so that
the optimization would be possible. On the other hand, if we are to
preserve the appropriateness of the estimate, the density can not be
too smooth. We calculate the expected value of the process by
numerical integrating of the marginal density of the SDE.
17
Table 1: The estimate of the parameters of cubic drift and constant
diffusion for the first three principal components of the Czech YC. In
addition, the constant c from the formula (7).
1PC 2PC 3PC
0α 6.69E-05 0.000257 2.99E-05
1α -0.00028 -0.00081 -0.00306
2α -0.00016 -0.00412 0.001473
3α -0.00039 -0.0045 -0.11237
0β 0.001914 0.000129 3.02E-05
c 1.82 1.82 1.06
From the bottom chart in figure 4, we can observe strong non-normality
and multimodality of the marginal density of 1PC
. The upper left chart
displays the cubic drift. It is obvious, that the component 1PC
(interpreted as the parallel shift of the YC) has the drift very close to
zero in interval ( 4− , 4 ). That means that the process behaves
similarly as random walk in this interval. This can be seen from the
chart of daily differences of the time series of 1PC
. The values are
more or less symmetrical around zero. Outside this interval, the
influence of the drift starts to increase very fast and it "pulls" the
process back strongly into the "non stationary interval". This behavior
of the process guaranties global stationarity of 1PC
. Similar results
were presented in the article [4] by Yacine-Ait Sahalia for American
one month Eurodollar rate.
The bottom charts of figures 5 and 6 show that the marginal densities
for the components 2PC
and 3PC
seem to be unimodal with a very
high kurtosis. The drift of both of these processes is again mean
reverting, and again the mean reverting property is relatively weak
around the value zero (i.e. around the expected value of the
processes) and its influence increases with the distance from zero. So
the processes of 2PC
and 3PC
are globally stationary as well.
The shape of the drifts of 1PC
, 2PC
and 3PC
implies the mean
reverting property for the interest rates. Global stationarity of these
processes supports the opinion that on a long period the process of the
YC is stationary as well. That means the rejection of the unit roots and
stability of the solution of the difference equations of the processes in
the methodology of linear discrete models.
18
5. SIMULATION
5.1. Simulation of the yield curve
The main purpose of our model is to quantify the dynamics of all
considered maturities of the yield curve (YC). We use simulations to
get possible realizations of the process of the YC. Simulations are
widely used, e.g. in the Monte Carlo methods of interest rate risk
management. Simulations can form the inputs for VaR calculation or
for portfolio optimization with respect to its future expected revenues,
costs and risk. Simulations can be used for profitability and risk
evaluations of derivative instruments and market strategies.
We simulate the yield curve process simulating the process for the first
three principal components of the YC. We use the Euler discretization
for the processes of the principal components. The Euler discretization
of SDE with cubic drift and constant diffusion (formula (5)) is
,)(= 0
3
3
2
210 ttttttt ttxxxxx εβαααα ∆+∆++++∆+ (13)
where
(0,1)Nt :ε . It is desirable to set t∆ as small as possible. The
results presented in this article are calculated with 1/100=t∆ .
Parameters of the model are estimated using daily observations of the
first three principal components of logarithms of the YC. Setting
1/100=t∆ means that there are 99 values generated between today
(timet ) and tomorrow (time 1+t ). Because of technical difficulties we
saved daily simulations only till the end of 2011. The beginning of the
simulation is set to be 19.10.2010. Each of the 295 saved values will
be referred to as a simulation step. We perform 10 000 simulations for
each of the three principal components.
For each simulation, we calculate the values of the YC "composing"
the principal components back in to the original process. The
generated values for a particular maturity then present one possible
path of its process. Altogether we get 10 000 paths for each maturity
for the next year.
5.2. Simulation evaluation
For practical purposes, we need to check whether the model is
appropriate. We verify the appropriateness of the model by comparing
the statistics of the historical YC (used for the model estimation) with
the statistics of the simulations. We presume that the similarity of the
statistics of the historical data and the simulations verifies correct
parametrization of the model.
19
To investigate the real properties of processes, the horizon of thirty
years would be appropriately long. The mean reverting property can
sometimes be observable only in a very long period. If we want to
explore the properties of the simulated processes, such a long horizon
is required. In some cases (state debt portfolio analysis, pension funds
portfolio analysis etc.), this horizon can have practical usage as well.
Unfortunately, the thirty-year horizon is too demanding
computationally. We use only the 14-months horizon of simulations in
our evaluation.
5.2.1. Longitudinal statistics
First we consider statistics that characterize the development of an YC
in a particular time period. We will refer to these statistics as to
longitudinal statistics. The time series of interest rates that were used
to estimate the parameters will be referred to as historical interest
rates.
We estimate the equilibrium of an interest rate with a simple average of
the historical interest rate. We request the model to revert to this
equilibrium (we request the mean reversion property). That means that
the averages of simulations of an interest rate calculated in each
simulation step over all simulations should converge to an average
historical interest rate in long time horizon.
Next we compare the equilibrium (historical average) with the average
of longitudinal averages calculated on a period of 14 months of the
simulation of the interest rates. We define the statistics with the
following formulas:
,
1
=
1=
ττ
i
n
i
h x
n
A ∑
(14)
sjwherex
t
A ij
t
i
j ,1,=,
1
=
1=
Kττ
∑
(15)
,
11
=
1
=
1=1=1=
τττ
ij
t
i
s
j
j
s
j
s x
ts
A
s
A ∑∑∑
(16)
where
τ
hA
is the equilibrium of the interest rate (the mean of the
historical interest rate),
τ
jA
is longitudinal average of a particular
simulation,
τ
sA
is average of longitudinal averages of all simulations, x
is the interest rate, τ is the maturity, n is the number of observations
of the historical interest rate, t is the count of simulations and s is the
count of simulations. The values of
τ
hA
and
τ
sA
as well as 2,5% and
20
97,5% percentile for
τ
jA
for all considered maturities are displayed in
table 2.
Table 2: Comparison of longitudinal averages and longitudinal
standard deviations.
τ
hA τ
sA 2,5% 97,5%
3M 2.95 1.52 1.13 2.10
6M 3.04 1.72 1.32 2.32
9M 3.12 1.84 1.43 2.46
1Y 3.20 1.95 1.53 2.58
3Y 3.62 2.28 1.84 2.91
5Y 3.95 2.56 2.09 3.19
10Y 4.43 2.97 2.48 3.62
15Y 4.67 3.24 2.74 3.88
Table 3: Comparison of longitudinal averages and longitudinal
standard deviations.
τ
hSD τ
sSD 2,5% 97,5%
3M 1.19 0.22 0.08 0.54
6M 1.17 0.21 0.08 0.52
9M 1.16 0.21 0.09 0.51
1Y 1.15 0.21 0.09 0.50
3Y 1.14 0.22 0.09 0.52
5Y 1.14 0.24 0.10 0.56
10Y 1.10 0.26 0.10 0.59
15Y 1.04 0.27 0.10 0.59
Similarly, we compare the longitudinal standard deviations. We define
longitudinal standard deviations analogically to longitudinal averages:
,)(
1
1
= 2
1=
ττ
hi
n
i
h Ax
n
SD −
− ∑
(17)
sjwhereAx
t
SD jij
t
i
j ,1,=,)(
1
1
= 2
1=
Kττ
−
− ∑
(18)
,)(
1
11
=
1
= 2
1=1=1=
τττ
jij
t
i
s
j
j
s
j
s Ax
ts
SD
s
SD −
− ∑∑∑
(19)
21
where
τ
hSD
is the standard deviation of the historical interest rate and
τ
sSD
is the average of standard deviations of all simulations. The
values of
τ
hSD
and
τ
sSD
as well as 2,5% and 97,5% percentiles of
longitudinal standard deviations of simulations are displayed in table 3.
The last considered longitudinal statistic is the correlation matrix of the
YC, i.e. the estimate of the correlations between the considered
maturities. The historical correlation matrix is calculated from the time
series of the interest rates. The correlation matrices of the simulations
of YC are (as in the case of longitudinal averages and standard
deviations) calculated from all simulations. The historical correlation
matrix, the matrix of average correlations of the simulations as well as
the matrix of 2,5% and 97,5% percentiles of correlations of the
simulations are displayed in table 4, 5, 6, 7 respectively.
Table 4: Comparison of longitudinal correlations.
sim 3M 6M 9M 1Y 3Y 5Y 10Y 15Y
3M 1.00 1.00 0.99 0.99 0.92 0.89 0.85 0.83
6M 1.00 1.00 1.00 1.00 0.93 0.89 0.85 0.83
9M 0.99 1.00 1.00 1.00 0.94 0.90 0.85 0.83
1Y 0.99 1.00 1.00 1.00 0.94 0.90 0.85 0.83
3Y 0.92 0.93 0.94 0.94 1.00 0.99 0.96 0.94
5Y 0.89 0.89 0.90 0.90 0.99 1.00 0.99 0.98
10Y 0.85 0.85 0.85 0.85 0.96 0.99 1.00 1.00
15Y 0.83 0.83 0.83 0.83 0.94 0.98 1.00 1.00
Table 5: Average correlations
sim 3M 6M 9M 1Y 3Y 5Y 10Y 15Y
3M 1.00 0.98 0.95 0.92 0.85 0.82 0.80 0.78
6M 0.98 1.00 0.99 0.97 0.88 0.83 0.77 0.74
9M 0.95 0.99 1.00 0.99 0.89 0.82 0.74 0.71
1Y 0.92 0.97 0.99 1.00 0.89 0.81 0.71 0.68
3Y 0.85 0.88 0.89 0.89 1.00 0.98 0.91 0.88
5Y 0.82 0.83 0.82 0.81 0.98 1.00 0.97 0.95
10Y 0.80 0.77 0.74 0.71 0.91 0.97 1.00 1.00
15Y 0.78 0.74 0.71 0.68 0.88 0.95 1.00 1.00
22
Table 6: Correlations of 2.5% percentile of simulations
sim 3M 6M 9M 1Y 3Y 5Y 10Y 15Y
3M 1.00 0.91 0.78 0.64 0.80 0.75 0.60 0.59
6M 0.91 1.00 0.96 0.89 0.45 0.81 0.62 0.62
9M 0.78 0.96 1.00 0.98 0.51 0.88 0.55 0.65
1Y 0.64 0.89 0.98 1.00 0.51 0.90 0.50 0.62
3Y 0.31 0.45 0.51 0.51 1.00 0.95 0.57 0.72
5Y 0.22 0.26 0.26 0.23 0.90 1.00 0.84 0.76
10Y 0.17 0.08 0.00 -0.06 0.57 0.84 1.00 0.99
15Y 0.11 -0.01 -0.09 -0.15 0.46 0.76 0.99 1.00
Table 7: Correlations of 97.5% percentile of simulations
sim 3M 6M 9M 1Y 3Y 5Y 10Y 15Y
3M 1.00 1.00 1.00 1.00 0.99 0.99 0.99 0.98
6M 1.00 1.00 1.00 1.00 0.99 0.99 0.98 0.98
9M 1.00 1.00 1.00 1.00 0.99 0.99 0.98 0.98
1Y 1.00 1.00 1.00 1.00 0.99 0.99 0.98 0.98
3Y 0.99 0.99 0.99 0.99 1.00 1.00 0.99 0.99
5Y 0.99 0.99 0.99 0.99 1.00 1.00 1.00 1.00
10Y 0.99 0.98 0.98 0.98 0.99 1.00 1.00 1.00
15Y 0.98 0.98 0.98 0.98 0.99 1.00 1.00 1.00
5.2.2. Unconditional (marginal) statistics
The shape of the drift estimated above implies that the process is
globally stationary. That means that the probability that the interest rate
will be e.g. 5 % is the same now as in 50 or 100 years. Time series of
a yield curve that is long enough is a sample from marginal distribution
of the stochastic process of the YC. For an appropriate model the
marginal statistics should be approximately equal to the historical ones
in the long run.
For the comparison of unconditional statistics, we will use analogical
measures as in the case of longitudinal statistics. The historical interest
rate average
τ
hA
is defined with the formula (14). We estimate the
unconditional average of simulations
τ
TsA , as the average of the
generated values in the very last simulation step:
23
,
1
=
1=
,
ττ
jT
s
j
Ts x
s
A ∑
(20)
where jTx
is the value of j -th simulation in the last simulated time
tickT , s is the count of simulations and τ is the maturity. The results
are displayed in table 8. The table 8 contains also the 95 % confidence
interval (calculated using the Student's t -distribution) for the mean of
the simulations. The confidence interval should cover the historical
average again in the long run.
Table 8: Comparison of the marginal averages
τ
hA τ
TsA ,
2,5% CI 97,5% CI
3M 2.95 1.74 1.65 1.89
6M 3.04 1.90 1.70 2,02
9M 3.12 2.01 1.85 2.08
1Y 3.20 2.10 2.01 2.25
3Y 3.62 2.47 2.38 2.62
5Y 3.95 2.78 2.69 2.90
10Y 4.43 3.25 3.05 3.42
15Y 4.67 3.53 3.33 3.75
Table 9: Comparison of the marginal standard deviations
τ
hA τ
TsA ,
2,5% CI 97,5% CI
3M 1.19 0.47 0.40 0.52
6M 1.17 0.47 0.40 0.52
9M 1.16 0.48 0.42 0.53
1Y 1.15 0.49 0.43 0.55
3Y 1.14 0.49 0.49 0.56
5Y 1.14 0.51 0.45 0.58
10Y 1.10 0.53 0.44 0.59
15Y 1.04 0.53 0.43 0.62
Similarly to the case of the unconditional average, we compare the
unconditional standard deviations and the unconditional correlation
matrices. The unconditional standard deviations of the simulations as
well as the confidence intervals (calculated from
2
χ distribution) are
displayed in table 9. The comparison of the correlation matrices is
displayed tables 10 and 11.
24
Table 10: Comparison of the marginal correlation matrices - historical
correlations
3M 6M 9M 1Y 3Y 5Y 10Y 15Y
3M 1.00 0.99 0.98 0.97 0.85 0.79 0.72 0.69
6M 0.99 1.00 1.00 0.99 0.87 0.79 0.71 0.67
9M 0.96 0.95 0.95 0.94 0.88 0.79 0.85 0.67
1Y 0.97 0.99 1.00 1.00 0.88 0.80 0.70 0.85
3Y 0.90 0.87 0.88 0.88 1.00 0.99 0.94 0.92
5Y 0.78 0.79 0.79 0.80 0.95 1.00 0.98 0.97
10Y 0.72 0.71 0.71 0.70 1.00 0.98 1.00 1.00
Table 11: Comparison of the marginal correlation matrices correlations
of the simulations in the time T (mean)
sim 3M 6M 9M 1Y 3Y 5Y 10Y 15Y
3M 1.00 0.99 0.98 0.97 0.85 0.79 0.72 0.69
6M 0.99 1.00 1.00 0.99 0.87 0.79 0.71 0.67
9M 0.98 1.00 1.00 1.00 0.88 0.79 0.71 0.67
1Y 0.97 0.99 1.00 1.00 0.88 0.80 0.70 0.66
3Y 0.85 0.87 0.88 0.88 1.00 0.99 0.94 0.92
5Y 0.79 0.79 0.79 0.80 0.99 1.00 0.98 0.97
10Y 0.72 0.71 0.71 0.70 0.94 0.98 0.92 0.99
15Y 0.69 0.85 0.67 0.69 0.96 0.97 1.00 0.98
6. CONCLUSIONS
We proposed a model of the yield curve of the Czech interest rate
market. We reduced the dimension of the process of the YC using the
principal components analysis. That way we can quantify the dynamics
of the process of the YC using models only for the first three principal
components. We used stochastic differential equations. We estimate
the parameters of the processes minimizing the difference between
parametric and nonparametric estimate of its marginal densities. That
way, we avoided the bias that occurs in common approaches based on
discretization of the SDE. This model has practical application in
interest rate risk management, such as debt management offices,
pension funds, mutual funds, hedge funds, etc.
This model is implemented and tested by the State Debt and Financial
Assets Department at Ministry of Finance of the Czech Republic. This
model should be used for prediction of interest cost of state debt.
25
7. REFERENCES
[1] CICHA M., KLADIVKO K. and ZIMMERMANN P.: Yield Curve
Modeling using Principal Component Analysis and Nonlinear
Stochastic Differential Equations, in Mathematical Methods in
Economics and Industry, Herlany, 2007. ISBN: 978-80-89089-58-1
[2] KARLIN, S. and TAYLOR, H. M.: A second course in Stochastic
Processes, Academic Press, 1981. ISBN-13: 978-0123986504
[3] LITTERMAN, R. and SCHEINKMAN: Common Factors affecting
bond returns, Journal of Fixed Income, 1, 54-61, 1991 ISSN: 1059-
8596
[4] SAHALIA, Y. A.: Testing Continuous-Time Models of the Spot
Interest Rate. The Review of Financial Studies, 1996, Vol. 9, No. 2, p.
385-425, ISSN 0893-9454.
[5] SILVERMAN, B. W.: Density Estimation for Statistics and Data
Analysis. London: Chapman and Hall, 1986. ISBN-13: 978-
0412246203.