DOI: 10.5817/FAI2013-1-1 No. 1/2013
5
RISK PREMIA IN THE CZECH MONEY MARKET
Martin Pohl
University of Economics, Prague
Zlonicka 703/2, Prague, Czech Republic
E-mail: pmartin@centrum.cz
Abstract: We estimate risk premia in the Czech money market and we pay
special attention to the 2008-2009 crisis period. Our results imply
a rising forward premium and we argue that the error correction model is
the most appropriate method, but median may be used as a first guess estimator.
We estimated the term premium between the policy rate and various money
market interest rates. In this context, ARCH models proved to be useful
in reflection of non-stationarity observed in the data. The financial crisis caused
a structural break in our data sample, but the impact on the forward premium
was only brief and forward premia normalized quickly. The widening of the term
premium proved to be much more persistent, although it declined significantly
since the peak of the crisis.
Keywords: financial crisis, expectations hypothesis, money market, PRIBOR,
forwards
JEL Classification: E43, C32
Introduction
Money market interest rates serve as benchmarks for a significant portion
of loans, bonds and derivatives (IMF, 2008). They are a key component
of the monetary policy transmission channel. Disruptions, therefore, have a large
impact on other segments of financial markets, the financial system, the economy
and the effectiveness of monetary policy. Money markets were also the epicentre
of the 2008/2009 financial crisis and have received a lot of attention since then.
There are several studies that deal with money markets in developed countries,
but only few of them deal with the crisis in the context of emerging markets.
In our paper, we focus on Czech money markets and we pay special attention
to the 2008/2009 crisis period. Our starting point is the expectation hypothesis
of the term structure of interest rates. The expectation hypothesis offers
an intuitive framework that forms the core of most standard interest rate
structure models and is applied by practitioners in their investment decisions,
although its empirical support is not unanimous in literature (Gravelle, 1998
or Kotlán, 1999). Using this framework, we estimate the forward premium
(the difference between the expected market rate and corresponding forward rate
Financial Assets and Investing
6
agreement) and the term premium (the difference between the policy rate
and key money market rates). Also, we test whether the premia changed during
the crisis.
1 Biased expectation hypothesis
The starting point of our analysis is the biased expectation theory and we offer its
brief overview in this section. There are more versions of expectation theory,
but in the context of our paper the forward premium and term premium concepts
are relevant (Costa, 2007). Biased expectation theory assumes that forward
interest rates reflect the expected path of short-term interest rates and risk
premia. In applying expectation theory restrictions, we may decompose
the forward rate ),( khh
tf +
into the expected short-term interest )( k
htt yE +
and forward premium f
tfp , where k denotes k-period instrument and h means
“h-months-ahead” horizon.
)(),( k
htt
f
t
khh
t yEfpf +
+
+= (1)
Standard expectation theory implies that a forward premium is higher
for longer-dated instruments that bear more interest rate risks for the investor.
We assume a constant forward premium which is a standard approach. We try
to derive market expectations from the actual development of interest rates
after removing random noise, assuming a constant forward premium. Equation
(1) may then be transformed into equation (2) for three-month interbank
deposits represented by variable
3
hty + , which builds the base of our empirical
analysis.
tht
hh
t yf εβα ++= +
+ 3)3,(
* (2)
A term premium is a premium based on the yield to maturity hypothesis that
assumes that yield on a zero-coupon bond
m
ty (deposits) that matures in m
periods (months) is equal to the average of one-period yields. In this context,
term premium
m
ttp represents excess return over rolling over one-period bonds
(deposits). We use linear approximation, but we agree that using compounded
returns would be more precise.
)(*
1
1
1
1∑=
−++=
m
i
it
m
t
m
t yE
m
tpy (3)
We would expect both forward and term premium to increase with rising
maturity. We assume them to be constant in our paper, but we discuss
the impact of structural breaks. In our analysis, we compare the CNB’s two-week
No. 1/2013
7
repo rate and various deposit rates. The repo rate is the leading interest rate
for market players and spreads below corresponding two-week deposits are
pretty low, averaging 5-10bp over most of the sample period, although they
widened significantly during the crisis period.
2 Methodology and data
We estimated the forward and term premium using different methods. We applied
cointegration analysis, forward regression, ARCH model and basic descriptive
statistics. In the Czech context, a similar approach was pplied by Kotlán (1999).
In contrast to him, we applied an error correction model that was employed
by other researchers analysing foreign money markets. Early works applying
cointegration technique similar to our approach may be found in Gravelle, Muller,
Stréliski (1998) who deal with Canadian money markets. For Euro markets
we found Costa, Galvão (2007) and Durré, Snorre, Pilegaard (2003) useful,
although they deal with implied forwards. A comprehensive comparison
of different instruments for US markets is provided by Gürkaynak, Sack, Swanson
(2007).
Our data set includes three month PRIBOR and corresponding forward rate
agreements (FRA) starting in one, three, six and nine months. Three month
PRIBOR rates and other money market rates are from the CNB. FRA rates (ask)
quotations and data used for illustrative purposes in the financial crisis section
are from Bloomberg. We use monthly averages with a sample start in January
2000. There are three reasons for this choice. Firstly, the data in 1997 and 1998
were impacted by the 1997 financial crisis, when interest rates reached high
double digit levels. Secondly, the Euro was introduced in 1999 and also the CNB
had a one-year experience with inflation targeting. Both events could have
caused structural breaks in the data sample. The sample ends in mid-2010,
when the consequences of the last financial crisis continued to fade away.
The relatively high level of interest rates (see Figure 1) at the beginning
of the sample was caused by the financial crisis in 1997 and the CNB’s reaction
to the crisis. The subsequent decline reflects remarkable disinflation and maturing
of the Czech economy and financial markets, which pressed Czech interest rates
to levels comparable to mature markets. This trend was interrupted by two
monetary tightening cycles and by the financial crisis, which impacted money
markets in 2008/2009.
Financial Assets and Investing
8
Figure 1 Selected money market interest rates
Source: CNB
Descriptive statistics for key variables are provided in Table 1. All data were
calculated as monthly averages. Our data sample shows important characteristics
that we study further in our paper. Firstly, we may observe that the interest rate
level rises with maturity. This is consistent with an upward sloping yield curve,
with risk premia rising depending on maturity. Secondly, the volatility seems to
be slightly higher for shortest maturities, although less visible for FRAs. Thirdly,
all distributions are skewed towards higher values as indicated by positive
skewness and a lower median than average. This is a typical picture, but may
even be reinforced by the impact of 2008/2009 financial crisis.
Table 1 Descriptive statistics (part I) – money market interest rates in %
Statistics cnb MM 1M MM 2M MM 3M MM 6M MM 9M MM 12M
Mean 2.94 3.06 3.10 3.15 3.24 3.32 3.40
Median 2.50 2.53 2.56 2.61 2.69 2.80 2.91
Maximum 5.25 5.40 5.48 5.57 5.73 5.85 5.98
Minimum 0.75 1.01 1.11 1.24 1.52 1.66 1.75
Std. Dev. 1.32 1.30 1.28 1.27 1.26 1.26 1.26
Skewness 0.57 0.60 0.60 0.61 0.65 0.67 0.69
Kurtosis 2.15 2.06 2.02 1.99 2.03 2.10 2.18
Source: author's estimate, Czech National Bank
Interest rates are assumed to be stationary, but in finite samples they often show
non-stationary behavior. We used standard tests to verify non-stationary data.
Individual unit root tests give weak support to the unit root hypothesis and
results are provided in Table 2.
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
00 01 02 03 04 05 06 07 08 09 10
CNB PRIBOR 3M PRIBOR 12M
No. 1/2013
9
Table 1 Descriptive statistics (part II) – FRAs in %
Statistics FRA_1to4 FRA_3to6 FRA_6to9 FRA_9to12
Mean 3.24 3.22 3.32 3.47
Median 2.70 2.78 2.88 3.03
Maximum 6.60 5.86 6.03 6.66
Minimum 1.25 1.20 1.27 1.38
Std. Dev. 1.33 1.27 1.29 1.33
Skewness 0.60 0.65 0.70 0.76
Kurtosis 2.05 2.21 2.32 2.50
Source: author's estimate
Table 2 Individual unit root tests (part I) – money market interest rates
Test cnb MM 1M MM 2M MM 3M MM 6M MM 9M MM 12M
PP -1.90 -2.03 -1.98 -1.92 -1.90 -1.88 -1.89
p-value 0.05 0.04 0.05 0.05 0.05 0.06 0.06
ADF -1.50 -2.09 -1.79 -1.68 -1.60 -1.79 -1.81
p-value 0.12 0.04 0.07 0.09 0.10 0.07 0.07
Legend: PP denotes Phillips-Perron test and ADF Augmented Dickey-Fuller test.
Source: author's estimate
Table 2 Individual unit root tests (part II) – FRAs
Test FRA_1to4 FRA_3to6 FRA_6to9 FRA_9to12
PP -1.59 -1.92 -1.82 -1.69
p-value 0.10 0.05 0.07 0.09
ADF -1.63 -1.87 -1.79 -1.77
p-value 0.10 0.06 0.07 0.07
Source: author's estimate
Group unit root tests are recommended for their higher power. They provide
more conclusive support for unit roots. We divided our sample into two
subsamples. The first subsample includes money market interest rates and policy
rate. The second subsample includes FRAs and three month PRIBOR. The results
are summarized in Table 2. All tests support the hypothesis that we deal with
non-stationary data. On the other hand, we acknowledge that unit root tests are
problematic from the methodological point of view and highly dependent
on sample choice.
Financial Assets and Investing
10
Table 3 Group unit root tests (part I) – money market interest rates
Test Statistic Probability
Levin, Lin & Chu t* -0.78 0.22
Im, Pesaran and Shin W-stat 0.18 0.57
ADF - Fisher Chi-square 6.08 0.81
PP - Fisher Chi-square 5.54 0.85
Source: author's estimate
Table 3 Group unit root tests (part II) – FRAs
Test Statistic Probability
Levin, Lin & Chu t* -0.81 0.21
Im, Pesaran and Shin W-stat 0.43 0.67
ADF - Fisher Chi-square 7.71 0.90
PP - Fisher Chi-square 7.84 0.90
Source: author's estimate
3 Forward premium
We decided to restrict our empirical tests on FRAs. In the absence of a futures
market, FRAs are the most reliable reflection of market expectations. We avoided
using implied forwards. Our experience shows that implied forwards are less
stable than FRAs and results are model dependent. This is especially true for less
liquid markets.
Descriptive statistics
The first useful way to look at the forward premium is to compare forward rates
with actual interest rates in the corresponding period. Basic descriptive statistics
are provided in Table 4.
The data support expectation of a rising time premium hypothesis. The spread
between actual 3M PRIBOR and 1 to 4 months FRA is not significant, but grows
strongly for more distant FRAs. The shape of the distribution changes depending
on the forward’s horizon. The distributions become flatter and skewed
toward higher spreads for more distant forwards. This reflects lower accuracy
of long-term expectations. Considering the shape of the distribution we think
that the median is better estimator of the forward premium than the average.
The median is less sensitive to the positive skewness of the distribution
and to occurrence of large observations.
No. 1/2013
11
Table 4 Difference between FRAs and actual 3M Pribor
Statistics DFRA_1TO4 DFRA_3TO6 DFRA_6TO9 DFRA_9TO12
Mean 0.05 0.15 0.35 0.60
Median 0.04 0.13 0.26 0.48
Maximum 0.46 1.08 1.73 2.32
Minimum -0.39 -0.64 -0.57 -1.04
Std. Dev. 0.14 0.32 0.51 0.71
Skewness -0.01 0.26 0.32 0.34
Kurtosis 3.96 3.23 2.50 2.53
Legend: Abbreviations denote spreads between FRAs and actual 3M PRIBOR, i. e.
DFRA_1to4 - spread between one to four month FRA and actual 3M PRIBOR etc.
Estimates correspond to percentage points.
Source: author's estimate
Forward regression
The standard regression expectation theory test is based on Equation (2) that is
transformed by subtracting the current level of interest rates (see Durre 2003)
and rearranging terms. This transformation should overcome the possible
non-stationarity of interest rates that could bias our estimates.
tt
hh
ttht yfyy εβα +−+=− +
+ )(*)( 3)3,(33
(4)
We estimate the equation on a horizon of 1, 3, 6 and 9 months corresponding to
FRAs. The validity of the unbiased expectation theory would be confirmed by
parameter values α=0 a β=1. A positive alfa parameter would imply a positive
forward premium. The results are provided in Table 5 with parameter estimates in
the first row and the t-test in the second. The third column includes R2
statistics.
All results were obtained by using the Newey-West corrected covariance matrix to
eliminate the bias caused by autocorrelated data.
The results support the expectations hypothesis. The beta parameter estimates
are slightly above one and Wald’s restriction test confirms that beta parameter
estimates are not significantly different from one for all equations. We also
imposed the restriction β=1 and estimated alfa parameters are in the last column
(with reverse signs). They are equal to the sample averages and seem to have an
upward bias as well. Models suffer from several weaknesses. Residuals are
correlated and show ARCH effects. The Chow’s and Quandt-Andrews breakpoint
stability tests show that parameters are unstable.
Financial Assets and Investing
12
Table 5 Forward regression
Maturity α β r2
αІβ=1
FRA_1to4 -0.05 1.06 0.33 0.05
-3.80 4.80 4.24
FRA_3to6 -0.15 1.02 0.34 0.15
-3.58 5.90 3.21
FRA_6to9 -0.37 1.11 0.35 0.35
-4.32 5.99 4.02
FRA_9to12 -0.63 1.11 0.31 0.60
-4.79 5.63 4.52
Legend: Abbreviations in lines denote FRAs of different maturity, i. e. FRA_1to4 - FRA
on 3M PRIBOR starting in one month. Parameter estimates and r2
statistics are
in columns with t-statistics in italics. Parameter covariance matrix estimated
by the Newey-West HAC method. Estimates correspond to percentage points.
Source: author's estimate
ARCH models
We concluded in the previous section that forward premium estimates are
probably biased and residuals show ARCH effects. Thus, we decided to include
the ARCH effect explicitly to remove the bias. We added a standard conditional
variance equation.
∑=
−+=
p
i
itth
1
2
0 εα (5)
The ARCH effect was not significant for one month ahead FRA; for longer horizons
we found only the first lag to be statistically significant. The results are shown in
Table 6. The support for β=1 hypothesis is weaker than in the constant variance
test, but it was not rejected for any model at a 5% significance level by the Wald
test. By imposing the β=1 restriction, we obtained risk premium estimates that
are significantly lower than in the previous case.
Table 6 Forward regression with ARCH effect
Maturity Mean Equation Variance Equation Restricted
α β α ARCH(1) α
FRA_1TO4 -0.05 0.92 0.01 0.24 0.05
-4.28 5.77 4.64 1.82 5.13
FRA_3TO6 -0.17 0.52 0.02 0.94 0.15
-12.25 4.78 3.88 5.85 10.61
FRA_6TO9 -0.25 0.80 0.05 0.81 0.27
-11.21 10.28 3.43 7.51 13.17
FRA_9TO12 -0.36 0.97 0.06 0.91 0.36
-12.40 19.25 3.73 8.21 15.73
Legend: Abbreviations in lines denote FRAs of different maturity, i. e. FRA_1to4 - FRA
on 3M PRIBOR starting in one month. Parameter estimates are in columns with tstatistics
in italics. Estimates correspond to percentage points.
Source: author's estimate
No. 1/2013
13
Cointegration analysis
Cointegration analysis has several advantages in comparison to the static
approach. Firstly, it is suited for non-stationary data. Secondly, it enables to test
causality and exogenity of variables. Thirdly, it interprets the relationship
of variables in terms of long-term equilibrium and and short-term dynamic.
On the other hand, there are also important weaknesses. Cointegration tests lack
robustness, results depend on sample choice and we are not able to effectively
deal with non-linearity often observed in the data.
We test the expectations theory by using the error correction framework. The
two-dimensional error correction model is described by the following set of
equations.
y
t
p
i
it
y
yy
i
a
p
i
hh
it
f
yf
i
a
t
y
hh
t
f
yt
y εµβα +∑
=
−
∆+∑
=
+
−
∆+−
−
−
+
−
=∆
1
3*
1
)3,(
*)3
1
*
)3,(
1
(*3 (6)
f
t
p
i
it
fy
i
p
i
hh
it
ff
it
hh
tf
hh
t yafayff εµβα +∆+∆+−−=∆ ∑∑ =
−
=
+
−−
+
−
+
1
3
1
)3,(3
1
)3,(
1
)3,(
**)*(* (7)
Endogenous variables 3M Pribor and FRA interest rates appear on both sides
of the equation. The error correction term is the first term on the right side
with error correction parameters alfa. The rest of the terms track the correlation
structure of variables. We choose the two-month lag length for all models
to remove linear dependence. The lag length seems to be appropriate for all
models with the exemption of 1 to 4 month forward, where the lag exclusion test
clearly supported only a one month lag. This is no surprise because both
instruments overlap for two months.
Our results are summarized in Table 7. The table starts with Johansen’s
cointegration test, where we report the trace statistics and corresponding
p-value. Johansen’s test confirms that a cointegration vector exists for all
forwards.
The second part of the table shows results of cointegration analysis. We imposed
restrictions on the beta parameter in line with expectation theory, what allowed
us to estimate the equilibrium forward premium directly from the cointegration
vector. The results are in the middle column. The forward premium rises for more
distant forwards and is statistically significant. The last column shows p-values for
the standard parameter restriction. Data for all models support the expectation
hypothesis that the parameter β is not significantly different from one which is in
line with expectation hypothesis. Error correction term parameters imply that
interest rates behave as expected. There is clear a Granger causality from
Financial Assets and Investing
14
forwards to spot interest rates. This means that deviations from the equilibrium
are corrected by spot rates. We also applied standard residual tests. The results
of the Lagrange multiplier test do not indicate residual autocorrelation,
but residuals suffer from ARCH effects and appear to be non-normal.
Table 7 Cointegration tests and vectors
Vector Johansen's test VEC
Error corr.
term
Restriction
H0 Trace p-value Premium MM 3M FRA p-value
MM 3M j=0 34.85* 0.00 0.04 0.75 0.36 0.99
FRA_1TO4 j<=1 3.96 0.42 4.13 3.88 1.42
MM 3M j=0 27.50* 0.00 0.10 0.32 0.16 0.68
FRA_3TO6 j<=1 3.55 0.48 3.50 4.60 1.40
MM 3M j=0 23.08* 0.02 0.23 0.21 0.09 0.73
FRA_6TO9 j<=1 3.91 0.43 4.98 4.77 1.18
MM 3M j=0 21.15* 0.04 0.40 0.15 0.02 0.80
FRA_9TO12 j<=1 4.21 0.38 5.73 4.35 0.38
Legend: Abbreviations in lines denote MM_3M for 3M PRIBOR and FRAs of different
maturity (FRA_1to4 - FRA on 3M PRIBOR starting in one month…). * we reject H0
hypothesis denoted by the number cointegration vectors, parameters t-statistics are
in italics. Estimates correspond to percentage points.
Source: author's estimate
Cointegration models are better suited to test the expectations hypothesis
than simply regression, but parameter values may not be constant especially
in the context of the last ten years. The financial crisis caused a clear structural
break in the development of many financial variables including money market
interest rates, including FRAs. The impact is clearly visible from the development
of the cointegration vector shown in Figure 2. Model residuals were exceptionally
large as well, but only for equations that included money market interest rates.
Figure 2 Cointegration vector
Source: author's estimate
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
00 01 02 03 04 05 06 07 08 09 10
FRA 3to6 FRA 6to9
FRA 9to12 FRA 1to4
No. 1/2013
15
The drop in the cointegration vector occurred in autumn 2008 and disappeared
quickly after February 2009. This may have impacted our results and we
estimated all models on a shorter sample ending in mid-2008. The results were
similar to full sample estimates and estimated parameter values differed by less
than 2bp. It seems that FRAs were relatively gently impacted by the 2008/2009
crisis with little evidence of long-lasting effects.
Conclusion
All estimates are summarized in Table 8. The highest estimates of the forward
premium were obtained by forward regression and mean statistics. We consider
these estimates biased and unreliable after theoretical and technical assumptions
were violated. We prefer the median as the first guess estimator, because the
statistic is robust to asymmetry observed in the empirical distribution. ARCH
models proved to be helpful in removing bias shown by forward regression, but
ARCH models largely ignore short-term dynamic that is driven by the monetary
policy cycle and correlated forecasting errors. Finally, we think that cointegration
framework offers best results. Error correction models enable to separate longterm
equilibrium and short-term dynamic. Our estimates are in line with the
biased expectations hypothesis and we also confirmed that forward premia
recovered relatively briskly from the financial crisis. On the other hand, all
methods assume the forward premium to be constant which may not be true.
Costa (2007) shows that the forward premium is time-varying and its size reflects
probability of capital losses for bond holders. This probability naturally increases
when interest rates are expected to go up and is usually accompanied by
positively skewed distribution of future interest rate changes. The asymmetric
probability of future changes is then reflected in positive skewness of distribution.
Relatively high estimates of the forward premium were also found for advanced
markets (see Gürkaynak 2007). There are two possible explanations. Firstly, the
forward premium also reflects the credit premium, but this is factor should be less
important in the derivate market in comparison to the term premium. Secondly,
investors may have overpredicted future interest rates in the past ten years that
were marked by declining interest rates and unexpected disinflation over the
period.
Financial Assets and Investing
16
Table 8 Forward premium estimates
Technique FRA_1to4 FRA_3to6 FRA_6to9 FRA_9to12
Mean 0.05 0.15 0.35 0.60
Median 0.04 0.13 0.26 0.48
Regression 0.05 0.15 0.35 0.60
ARCH 0.05 0.15 0.27 0.36
Cointegration 0.04 0.10 0.23 0.40
Legend: Abbreviations in columns denote FRAs of different maturity (FRA_1to4 FRA
on 3M PRIBOR starting in one month…). Estimates correspond to
percentage points.
Source: author's estimate
4 Term premium
The relation between policy rate and money market interest rates remains
the key point of interest for market participants and policy makers. We define
the term premium as the difference between the money market rate and policy
rate or as the excess return of a term deposit above rolling-over two-week repo
tenders. The key challenges for researchers are the impact of monetary policy,
possible structural breaks, nonlinearities or heteroskedasticity in the data. We do
not attempt to split the term premium into different categories because reliable
risk-free benchmarks are not available for Czech money markets.
Firstly, we look at simple descriptive statistics. Secondly, we apply cointegration
framework similarly to the forward premia section. Thirdly, we estimate ARCH
models to reflect the nonlinearity observed in the data.
Descriptive statistics
The first step to estimate the term premium is to use simple descriptive statistics.
We subtract the average policy rate from the corresponding money market rate
to derive spreads. Policy rate is represented by the two-week repo rate which is
a less risky instrument in comparison to the interbank deposit. We use linear
approximation with the following formal expression.
∑=
−+−=
m
i
it
m
t
m
t CNB
m
ys
1
1*
1
(8)
The results are summarized in Table 9. The spread of money market rates over
policy rises with maturity. The distribution of spreads has high kurtosis and is
skewed towards higher values. The mean may be biased upwards and so we
prefer the median. The distribution gets flatter and volatility larger with rising
maturity which reflects higher uncertainty of distant forwards. The weakness
No. 1/2013
17
of simple statistics is that we assume stable distribution and that it may be biased
by the presence of a relatively low number of tightening/easing periods in our
sample.
Table 9 Descriptive statistics
S_1M S_2M S_3M S_6M S_9M S_12M
Mean 0.11 0.17 0.24 0.37 0.48 0.59
Median 0.05 0.08 0.13 0.23 0.32 0.39
Maximum 1.07 1.37 1.66 2.18 2.44 2.66
Minimum -0.17 -0.08 -0.11 -0.12 -0.16 -0.23
Std. Dev. 0.18 0.24 0.31 0.42 0.52 0.60
Skewness 2.97 2.66 2.43 1.89 1.44 1.18
Kurtosis 14.24 11.30 9.99 7.23 5.20 4.16
Legend: Abbreviations in columns denote spreads between different money
market rates and actual two -week repo rate during the interbank deposits
lifetime (s_1M is spread between monthly averages of 1M PRIBOR and twoweek
repo rate). Estimates correspond to percentage points.
Source: author's estimate
Sample statistics were surely impacted by the financial crisis and Table 10
confirms it. Pre-crisis distribution of term spreads is less skewed, has lower
kurtosis and its mean is closer to the median. Both median and mean statistics
are lower than in the full sample estimates.
Table 10 Term premium statistics – pre-crisis sample
S_1M S_2M S_3M S_6M S_9M S_12M
Mean 0.04 0.06 0.10 0.20 0.30 0.41
Median 0.04 0.06 0.08 0.17 0.25 0.32
Maximum 0.15 0.25 0.33 0.69 1.06 1.47
Minimum -0.17 -0.08 -0.11 -0.12 -0.16 -0.23
Std. Dev. 0.04 0.05 0.08 0.19 0.30 0.39
Skewness -1.28 0.37 0.45 0.61 0.61 0.67
Kurtosis 9.01 4.65 3.60 2.81 2.59 2.69
Legend: Abbreviations in columns denote spreads between different money
market rates and actual two week repo rate during interbank deposits lifetime
(s_1M is spread between monthly averages of 1M PRIBOR and two week repo
rate). Estimates correspond to percentage points.
Source: author's estimate
Cointegration
Theoretically, cointegration is well-suited for estimates of term spread
in the money markets. Most importantly, it enables to decompose the relationship
between policy rates and different money market rates on a short-term
and long-term “equilibrium” part. On the other hand, we still assume
the “equilibrium” premium to be constant.
Financial Assets and Investing
18
The necessary condition for cointegration is non-stationarity of underlying
variables. The tests are represented in the appendix and they support
the non-stationarity hypothesis. Similarly to the forward premium section, we will
use an error correction framework that may be described by the following set
of equations.
y
t
p
i
m
it
yy
i
p
i
it
ycnb
i
m
tty
m
t yacnbaycnby εµβα +∆+∆+−−=∆ ∑∑ =
−
=
−−−
11
11 **)*(* (9)
cnb
t
p
i
m
it
cnby
i
p
i
it
cnbcnb
i
m
ttcnbt yacnbaycnbcnb εµβα +∆+∆+−−=∆ ∑∑ =
−
=
−−−
11
11 **)*(* (10)
The money market rate of maturity m and the average two-week repo rate
appear on both sides of the equations. The equilibrium or cointegrating equation
is described by µβ += m
tt ycnb * with the error correction parameter alfa. The rest
of the terms track the correlation structure of variables.
Cointegration tests failed to prove a cointegration relationship between policy rate
and money market rates, although the necessary condition was met
and fundamental logic also supports the expectation of a cointegration
relationship. We think that the structural break during the crisis period is the key
reason. We reduced the length of the sample to the January 2000 – June 2007
period. In this shorter sample, the data support cointegration hypothesis
decisively. Results are in Table 11.
All models were estimated with a three-month lag that was supported
by standard tests. The cointegration vector shows that term spread rises
with maturity and is statistically significant. The last column shows p-values
for parameter restriction. Data for all models support the hypothesis that
the parameter β is not significantly different from unity which is in line
with the expectation hypothesis. Only one-month maturity is an exemption.
The error correction parameters have expected sign, but only policy rate
parameters are statistically significant. Thus, disequilibrium is eliminated
by policy rates, which is consistent with expectation theory. Residuals seem to be
uncorrelated, but suffer from ARCH effects and are non-normal (money market
interest rate components).
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Table 11 Term premium estimates
Vector Johansen's test VEC
Error
correction
term
Restriction
H0 Trace p-value Premium MM cnb p-value
cnb j=0 62.7* 0.00 0.04 0.00 0.70 0.01
MM 1M j<=1 8.16 0.07 7.64 0.00 2.17 lag=3
cnb j=0 50.54* 0.00 0.06 0.08 0.72 0.17
MM 2M j<=1 3.60 0.47 6.85 0.41 4.04 lag=3
cnb j=0 40.17* 0.00 0.09 0.10 4.64 0.33
MM 3M j<=1 2.80 0.62 7.26 0.65 0.61 lag=3
cnb j=0 28.35* 0.00 0.17 0.03 0.36 0.54
MM 6M j<=1 3.58 0.48 7.53 0.22 4.09 lag=3
cnb j=0 26.81* 0.01 0.26 0.05 0.25 0.49
MM 9M j<=1 4.87 0.30 7.62 0.47 3.60 lag=3
cnb j=0 24.3* 0.01 0.34 0.06 0.18 0.48
MM 12M j<=1 2.88 0.20 7.44 0.68 3.28 lag=3
Legend:CNB denotes CNB's two-week repo rate. Other abbreviations in lines denote
money market rates of different maturity (MM 1M is one month PRIBOR).
Parameters t-statistics are in italics. * We reject H0 hypothesis denoted
by the number of cointegration vectors. Estimates correspond to percentage points.
Source: author's estimate
ARCH models
Money market interest rates exhibit heteroscadasticity with a structural break
in the crisis period. A standard approach on how to deal with heteroscadastic
data is to use GARCH models, although they are not well-suited to deal
with the structural break. We found a useful inspiration in Engle (1987), but we
decided to avoid ARCH-M specification, which includes the time-varying variance
in the mean equation. We find this approach misleading. The methodology is
not able to remove the impact of the monetary policy cycle from the data.
The ARCH-M model will then interpret the forecasting error as a part of the timevarying
risk premium.
We tested several variants of the GARCH(p,q) specification, but finally decided
to only use ARCH(p) models because “value added” of more complicated models
was very limited. We used term spread as dependent variable in the mean
equation and we assumed constant mean. Conditional variance equation
is depicted in equation (5). Our results are presented in Table 12.
For every maturity we present a model that minimized the Bayesian information
criterion.
Financial Assets and Investing
20
Table 12 Term premium estimates
Maturity
Mean Equation Variance Equation
Constant Constant ARCH(1) ARCH(2)
S_1M 0.05 0.00 1.53 -
13.10 2.67 3.77 S_2M
0.07 0.00 1.26 -
15.52 2.29 2.81 S_3M
0.10 0.00 0.76 0.43
21.58 3.05 2.84 2.19
S_6M 0.14 0.00 1.15 -
21.13 4.77 7.79 S_9M
0.19 0.02 0.95 -
14.79 3.05 9.52 S_12M
0.28 0.02 1.01 -0.08
19.68 3.85 11.25 -2.95
Legend: Abbreviations in rows denote spreads between different money
market rates and actual two-week repo rate during interbank deposits
lifetime (s_1M is spread between monthly averages of 1M PRIBOR
and two-week repo rate). Parameters z-statistics are in italic. Estimates
correspond to percentage points.
Source: author's estimate
Although ARCH models proved to be a useful tool to capture some sources
of non-stationarity in the data, they are not able to reflect the shift in the mean
that occurred during the financial crisis. The structural break is clearly visible
from the residual as shown in Figure 3.
Figure 3 Selected ARCH models residuals
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
00 01 02 03 04 05 06 07 08 09 10
1M 3M 6M 12M
Source: author's estimate
The shock was persistent, although it gradually faded away. The structural break
may have impacted our estimates of the term premium and therefore, we
estimated all models only for the pre-crisis period ending in June 2007 similarly
No. 1/2013
21
to the cointegration analysis. The premia estimates were similar to full sample
results, but ARCH effects became statistically insignificant. From this point
of view, ARCH models seem to be a robust technique in case we deal with
structural breaks and non-stationarity in the data.
A closer look at residuals also may give an idea of persistence of the structural
shock. Model residuals reached the maximum mostly in October 2010, when
the financial crisis also culminated in most segments of financial markets.
The elevated risk premia, as measured by residuals, started to gradually fade
away. Normalization was faster for shorter maturities, which is also in line
with evidence from foreign money markets.
Table 13 ARCH models - residuals in bp
Maturity -6M Peak
value
Peak
month
+6M Last
value
Last
month
S_1M 3 59 2008:11 11 4 2010:06
S_2M -6 65 2008:10 25 3 2010:05
S_3M 5 58 2008:10 14 18 2010:04
S_6M 3 68 2008:10 11 18 2010:01
S_9M 12 68 2008:10 9 24 2009:10
S_12M 20 69 2008:10 10 24 2009:07
Legend: Abbreviations in rows denote spreads between different money market
rates and actual two-week repo rate during interbank deposits lifetime (s_1M is
spread between monthly averages of 1M PRIBOR and two-week repo rate).
Source: author's estimate
Summary
Table 14 provides a summary of our estimates. We may conclude that the term
spread rises with maturity and that estimation methods differ in their sensitivity
to structural breaks and non-stationarity in the sample. Full sample estimates
that include the crisis period are higher for all methods and the mean is especially
sensitive to extreme observations. The ARCH model provides similar estimates
for the full sample and pre-crisis sample period, but the error correction model
gives useful estimates only for the pre-crisis period. Again, we believe that
the median may give a useful first guess estimate with other techniques giving
more precise estimates, but we should keep in mind that every method has
its weaknesses.
Financial Assets and Investing
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Table 14 Term premium – summary
Method MM 1M MM 2M MM 3M MM 6M MM 9M MM 12M
Pre-crisis sample (2000:01-2007:06)
Mean 0.04 0.06 0.10 0.20 0.30 0.41
Median 0.04 0.06 0.08 0.17 0.25 0.32
Cointegration 0.04 0.06 0.09 0.17 0.26 0.34
ARCH 0.04 0.07 0.10 0.13 0.19 0.26
Full sample (2000:01-2010:06)
Mean 0.11 0.17 0.24 0.37 0.48 0.59
Median 0.05 0.08 0.13 0.23 0.32 0.39
ARCH 0.05 0.07 0.10 0.14 0.19 0.28
Legend: Abbreviations in columns denote money market rates of different maturity
(MM 1M is one month PRIBOR). Estimates correspond to percentage points.
Source: author's estimate
5 Concluding remarks
The last ten years were remarkable for Czech money markets. Key interest rates
have normalized to levels typical in advanced economies. The financial crisis
provided considerable stress for all segments of financial markets including
money markets. Risk premia widened abruptly following the disintegration
of markets after Lehman Brother's collapse. Unfortunately, we lack a reliable
risk-free benchmark in the Czech money market and, therefore, it is impossible
to provide an accurate decomposition of money market rates into a risk-free part
and risk premium. Instead, we focused on key risk premia implied by
the expectation hypothesis framework – the forward premium and the term
premium. Our analysis showed that forward premia were relatively stable during
the financial crisis and they normalized quickly. The results generally support
the biased expectation hypothesis, but estimates differ significantly depending
on estimation methods. We argue that medians may provide the first guess
estimate of the forward premium, but ARCH models and cointegration techniques
are better suited to deal with non-stationarity observed in the data. We consider
the forward premium to be relatively high and we see two possible explanations.
Firstly, the forward premium also reflects the credit premium, but this factor
should be less important in the derivative market in comparison to the term
premium. Secondly, investors may have overpredicted the future interest rate
in the past ten years, marked by declining interest rates and unexpected
disinflation during the period. A similar conclusion can be drawn from the term
premium analysis, although the impact of the financial crisis was heavier. Our
estimates show a clear structural break during the crisis period, but elevated
premia considerably declined later. Methods that explicitly or implicitly assume
a stable term premium fail or provide biased estimates. Again, medians seem to
No. 1/2013
23
be a relatively robust estimator that less sensitive to nonlinearities and parameter
instability observed in the data. We found ARCH models to be useful as timedependent
variance enables to absorb a significant part of the shock that
occurred during the financial crisis.
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