This paper tries to find out, whether the Covered Interest Rate Parity (CIRP) theory was valid for exchange rate CZK/EUR during the period ranging from May 2001 to November 2007. As a main tool, a common OLS regression was chosen. It was augmented by MA(1) process of residuals and by ARCH (6) model of residuals’ variance. The results show, that the CIRP theory was not valid during selected period. However, it seems apparent, that the main factors for 3-month forward exchange rate CZK/EUR determination were an interest rate differential and a nominal spot exchange rate. This is fully consistent with the CIRP theory.Covered interest rate parity, exchange rate, interest rate, foreign exchange markets

Bednarik, Radek

English

Research Papers in Economics

COVERED INTEREST RATE PARITY: THE CASE OF THE CZECH 
REPUBLIC 
 
RADEK BEDNAŘÍK 
Ekonomická fakulta, VŠB-TU Ostrava 
 
 
Abstrakt 
Tento příspěvek se zabývá teorií kryté úrokové parity a snaží se zjistit, zda tato teorie platila 
pro devizový kurz CZK/EUR v období od května 2001 do listopadu 2007. Byla použita běžná 
OLS  regrese  doplněná  MA(1)  procesem  reziduí  a  ARCH(6)  modelem  rozptylu  reziduí
1. 
Výsledky ukazují, že teorie kryté úrokové parity v daném období neplatila, nicméně zdá se, že 
hlavními  faktory pro  formování  tříměsíčního  forwardového  kurzu  CZK/EUR byl  úrokový 
diferenciál a nominální spot kurz, což je plně v souladu s touto teorií. 
 
Klíčová slova 
Krytá úroková parita, devizový kurz, úroková míra, měnové trhy. 
 
Abstract 
This paper tries to find out, whether the Covered Interest Rate Parity (CIRP) theory was valid 
for exchange rate CZK/EUR during the period ranging from May 2001 to November 2007. As 
a main tool, a common OLS regression was chosen. It was augmented by MA(1) process of 
residuals and by ARCH (6) model of residuals’ variance. The results show, that the CIRP 
theory was not valid during selected period. However, it seems apparent, that the main factors 
for 3-month forward exchange rate CZK/EUR determination were an interest rate differential 
and a nominal spot exchange rate. This is fully consistent with the CIRP theory. 
 
Key words 
Covered interest parity, exchange rate, interest rate, foreign exchange markets. 
 
 
1  Introduction 
The  Covered  Interest  Rate  Parity  theory  (CIRP)  is  one  of  many  theories,  which  try  to 
formulate equilibrium conditions of foreign exchange markets. Its’ principles are simple. This 
theory posits that domestic assets’ interest rate should be equal to foreign assets’ interest rate 
plus  expected  percentual  change  of  exchange  rate.  This  is  in  fact  the  simple  form  of 
Uncovered Interest Rate Parity (UIRP). But, if we substitute expected future exchange rate 
with known future exchange rate (i.e. forward exchange rate), we get CIRP.  
The validity of CIRP in the Czech republic was tested, for example, by Komárková (2006), or 
by Komárek – Komárková (2008). She found out, that CIRP was valid in the Czech republic 
during examined period. However, Komárková (2006) uses OLS regression to test functional 
form of CIRP as it is given by equation (2). In this paper, we tests CIRP in slightly different 
way, using the equations (9) and (10). Nonetheless, our results also show that CIRP is valid, 
                                                 
1 A similar approach to account for autocorrelation of residuals is used in Stančík (2007). Although he uses 
AR(p) process, the principle is the same. but in terms of determining the forward exchange rate by spot exchange rate and interest 
rates’ differential.  
The aim of this paper is to test empirically, whether the CIRP holds for the nominal exchange 
rate CZK/EUR or not. We use daily data from May 2001 to November 2007. The econometric 
testing  incorporates  OLS  regression  augmented  by  MA(1)  process  of  residuals  and  by 
ARCH(6) model of residuals’ variance.  
The paper is divided into following sections. The first section defines CIRP, regressions’, 
MA(1) and ARCH(6) functions used for econometric testing. Also we display some graphs of 
data,  brief  description  and  some  conclusion  arising  from  them.  In  the  second  section  we 
present the results of econometric regression. The third section concludes. 
2  Creating the model of the Covered Interest Rate Parity 
2.1  The Covered Interest Rate Parity 
The CIRP model can be derived as follows. 
First, we have to specify the UIRP. The equation (1) defines the UIRP as equality between 
domestic and foreign assets’ interest rates ratio and ratio of future expected exchange rate and 
current  exchange  rate.  Assuming  that  the  future  value  of  foreign  assets  includes  a  risk 
premium, we can write
2: 
( )
( ) ( ) σ +
+
+
=
+ 1 *
1
1
* i
i
E
E
t
1 t ,  (1) 
where  Et+1  is  the  future  expected  exchange  rate, Et  is  the  current  exchange  rate, i  is  the 
domestic assets’ interest rate, i
* is the foreign assets’ interest rate and σ is the risk premium of 
foreign assets held by domestic agents due to exchange rate uncertainty. 
Assuming that foreign exchange markets are effective and therefore Ft is the best prediction 
of Et+1, we may substitute Et+1 with Ft, which is the known future exchange rate, i.e. the 
forward exchange rate and we get the equation of CIRP: 
( )
( )
* 1
1
i
i
E
F
t
t
+
+
= ,  (2) 
where Ft is forward exchange rate, Et is current (spot) exchange rate, i and i
* are domestic and 
foreign assets’ interest rates. Note that risk premium has been removed. It is simply because 
the uncertainty adherent to the possible expected exchange rate change has gone. When the 
equation (2) holds, the foreign exchange market of domestic and foreign currency is in the 
equilibrium state. 
                                                 
2 Or in another form: ( ) ( )
( ) t
t
E
E i
i
1
*
*
1
1
1
+
+
+
= +
σ
 2.2  Creating the OLS regression function 
If we want to apply the OLS regression on equation (2), we have to take the logarithms of 
selected equation. By doing this operation and after moving the logarithm of Et to the right 
side of the equation we get: 
( ) ( ) ( ) ( ) t t t E i i F ε + + + − + = ln 1 ln 1 ln ln
* ,  (3) 
or: 
t t t t t LS LIF LID LF ε β β β α + + + + = * * * 3 2 1 1 ,    (4) 
where LFt is natural logarithm of forward exchange rate, LIDt and LIFt are natural logarithms 
of  domestic  and  foreign  assets’  interest  rates.  LSt  is  natural  logarithm  of  spot  nominal 
exchange rate CZK/EUR and εt is the error term. α1 is constant and β1, β2, β3 are parameters, 
where β2 is assumed to be negative and β1 and β3 to be positive. Since testing for common unit 
root (Levin, Li, Chu; Breitung; and Hadri tests) shows that time series are integrated by order 
1, we have to use first differences.  
This yields in equation (5): 
t t t t t DLS DLIF DLID DLF ε β β β α + + + + = * * * 3 2 1 1 .  (5) 
This equation can be used for standard OLS regression. 
2.3  Defining MA(1) process of residuals 
Since OLS regression of equation (5) suffers from autocorrelation of residuals, to account for 
this undesirable effect we implement MA(1) process, which form is defined by equation (6): 
t t t υ γ φ ε + = − 1 1 ,  (6) 
where εt is value of residuals in time t, γt-1 is value of forecast error in time t-1 and υt is error 
term of equation (6). 
Thus, so far our regression equation has the following form: 
t t t t t t DLS DLIF DLID DLF υ γ φ β β β α + + + + + = − 1 1 3 2 1 1 * * * .  (7) 
2.4  Defining ARCH(6) models of residuals 
Since ARCH LM test of equation (7) shows that autoregressive conditional heteroskedasticity 
(ARCH) is present, we have to incorporate into our regression function a model of ARCH to 
capture this effect on variance of error term υt.  Presence of ARCH effect means that variance 
of residuals is not constant, which violates one of the conditions of valid OLS (including 
MA(1)) regression
3. We use ARCH(6) model, as it is specified below
4. 
                                                 
3 Implementation of ARCH model is also justified by the presence of volatility clustering in differenced time 
series data of spot and forward exchange rate. See Fig. 1. 2.4.1  ARCH(6) 
This model captures unstable variance of residuals by using following approach
5: 
∑
=
− + =
6
1
2 2 *
n
n t n t t υ α ω σ ,  (8) 
where σ
2
t is a variance of residuals υt, υ
2
t-n is a squared value of residual in time t-n. 
2.5  Final model for OLS estimation 
Our  model  which  was  used  for  OLS  regression  and  which  captures  the  effect  of 
autocorrelation and ARCH on residuals is presented by equations (9) and (10): 
t t t t t t DLS DLIF DLID DLF υ γ φ β β β α + + + + + = − 1 1 3 2 1 1 * * * ,  (9) 
∑
=
− + =
6
1
2 2 *
n
n t n t t υ α ω σ .  (10) 
Residuals υ are assumed to follow NID(µ,σ
2), where µ = 0 and σ
2 = 1. µ and σ
2  are supposed 
to remain constant during whole period, i.e. residuals are homoskedastic. 
2.6  Data 
For econometric testing of CIRP, daily data from May 2001 to November 2007 were used. 
For known future exchange rate (Ft) we use 3-month forward exchange rate of CZK/EUR, for 
current exchange rate (Et) we use spot exchange rate of CZK/EUR. As proxies of domestic 
and foreign interest rate (i and i
*), PRIBOR 3M and LIBOR 3M were chosen.  
Fig. 1 shows leveled and differenced logarithmic values of all used variables. 
                                                                                                                                                         
4 The ARCH (6) was chosen as the most suitable one after conducting several tests, which were aimed to pick 
the  best  model  of  ARCH  effect  of  residuals.  The  candidates  were  ARCH,  GARCH,  PARCH,  TARCH, 
EGARCH and Component ARCH models. The OLS regression which included MA(1) and ARCH(6) displayed 
the best results in terms of presence of no autocorrelation and of no remaining ARCH effect. 
5 See Arlt, J., Arltová, M. (2003, p. 166) or Arlt, J., Arltová, M. (2006, p. 128). Fig. 1 Development of leveled and differenced variables 
26
27
28
29
30
31
32
33
34
35
250 500 750 1000 1250 1500
FORWARD 3M
26
27
28
29
30
31
32
33
34
35
250 500 750 1000 1250 1500
SPOT CZK/EUR
1.01
1.02
1.03
1.04
1.05
1.06
250 500 750 1000 1250 1500
PRIBOR 3M
1.015
1.020
1.025
1.030
1.035
1.040
1.045
1.050
250 500 750 1000 1250 1500
LIBOR 3M
-.03
-.02
-.01
.00
.01
.02
250 500 750 1000 1250 1500
DIFF LN FORWARD 3M
-.004
-.003
-.002
-.001
.000
.001
250 500 750 1000 1250 1500
DIFF LN LIBOR 3M
-.005
-.004
-.003
-.002
-.001
.000
.001
.002
.003
250 500 750 1000 1250 1500
DIFF LN PRIBOR 3M
-.03
-.02
-.01
.00
.01
.02
250 500 750 1000 1250 1500
DIFF LN SPOT CZK/EUR
Source: CNB, BBA, own calculations
Note: The X axis is not time-labeled, since there were some unmatched gaps in time series.
To be able to match selected time series, some data have to be removed.
Hence, displayed data do not cover all days of selected period of time.
 
We  may  see  that  SPOT  CZK/EUR  and  FORWARD  3M  development  (in  leveled  or 
differenced form) is in fact almost the same. That is, of course, not surprising, because the 
forward exchange rate usually follows the same pattern as spot exchange rate does. Value of 
forward exchange rate differs from spot by value given by interest rate differential. Also from 
differenced  forms  of  forward  and  spot  exchange  rates  we  may  see  a  typical  pattern  of 
volatility clustering which justifies ARCH(6) implementation. 
 
Fig. 2 Comparison of Forward 3M/spot ratio and PRIBOR 3M/LIBOR 3M ratio 
0.985
0.990
0.995
1.000
1.005
1.010
1.015
1.020
250 500 750 1000 1250 1500
FORWARD 3M/SPOT PRIBOR 3M/LIBOR 3M
 
Given the equation (2), when CIRP holds, the values of these two ratios should be the same. 
Thus, seeing Fig. 2, we could conclude, that (strict) CIRP theory did not hold for CZK/EUR 
exchange  rate.  However,  as  we  will  see  in  the  2nd  section,  the  econometric  testing  of 
equations (9) and (10) shows us, that interest rate differential and spot exchange rate are the 
main components in determining known future exchange rate. Fig. 3 Relationships between regressand and regressors 
-.005
-.004
-.003
-.002
-.001
.000
.001
.002
.003
-.03 -.02 -.01 .00 .01 .02
DIFF LN FORWARD 3M
D
I
F
F
 
L
N
 
P
R
I
B
O
R
 
3
M
-.015
-.010
-.005
.000
.005
.010
.015
-.03 -.02 -.01 .00 .01 .02
DIFF LN FORWARD 3M
D
I
F
F
 
L
N
 
L
I
B
O
R
 
3
M
-.03
-.02
-.01
.00
.01
.02
-.03 -.02 -.01 .00 .01 .02
DIFF LN FORWARD 3M
D
I
F
F
 
L
N
 
S
P
O
T
 
C
Z
K
/
E
U
R
 
In Fig. 3 we can see that, there is clear linear relationship namely between forward and spot 
rate. That is, of course, not surprising, as it is apparent from Fig. 1. 
Fig. 4 Comparison of predicted spot rate by Forward 3M and real spot rate 
26
27
28
29
30
31
32
33
34
35
250 500 750 1000 1250 1500
FORWARD 3M SPOT AFTER 3 MONTHS
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
250 500 750 1000 1250 1500
F 3M AND SPOT BIAS IN ABSOLUTE VALUES
-4
-3
-2
-1
0
1
2
250 500 750 1000 1250 1500
FORWARD 3M AND SPOT BIAS
Source: CNB, own calculations
 
Looking at Fig. 4, we see that foreign exchange markets were not successful in predicting 
future spot exchange rate by forward 3M. We can see that during almost the whole examined 
period, the CZK/EUR rate was predicted by exchange markets to be weaker than it actually 
happened  to  be.  Also  we  may  notice  that  the  bias  of  predicted  and  real  spot  rate  was 
developing in cyclical pattern with decreasing range except for the end of examined period. 
3  Results of econometric testing of CIRP validity 
The model we test has form given by equations (9) and (10), i.e.: 
t t t t t t DLS DLIF DLID DLF υ γ φ β β β α + + + + + = − 1 1 3 2 1 1 * * * ,  (9) 
∑
=
− + =
6
1
2 2 *
n
n t n t t υ α ω σ .  (10) 
We test whether the 3-month forward exchange rate is determined by the 3-month domestic 
and foreign interest rate differential and by the spot exchange rate or not. The results are 
presented in Tab. 1. Tab. 1 Regression results
* 
Dependent Variable  DLF 
Independent Variables:   
Constant  0.0000 
(0.7289) 
DLID  0.2447 
(0.0000) 
DLIF  -0.2599 
(0.0000) 
DLS  1.0047 
(0.0000) 
MA(1)  -0.5107 
(0.0000) 
ARCH (6):   
Dependent Variable  σ
2
t 
Independent Variables:   
ωt  0.0000 
(0.0000) 
υ
2
t-1  0.1578 
(0.0002) 
υ
2
t-2  0.2194 
(0.0000) 
υ
2
t-3 
0.8639 
(0.0000) 
υ
2
t-4  -0.0754 
(0.0005) 
υ
2
t-5  0.1264 
(0.0000) 
υ
2
t-6  0.0926 
(0.0000) 
R
2  0.9982 
Adj. R
2  0.9982 
F-stat.  0.0000 
Autocorrelation:   
Correlogram  no autocorrelation 
Q-stat.  possible autocorrelation 
up to 24th lag. 
Normality:   
JB-test  not normal distribution 
(0.0000) 
No. of observations  1639 
* - probability values in parentheses. Results of tests of model’s validity 
are available upon request. 
From regression results we are able to clearly see, that interest rate differential and spot rate 
were crucial factors for forming forward exchange rates. 
The  results  of  autocorrelation  tests  are  mixed.  A  correlogram  suggests  that  there  is  no 
autocorrelation in residuals. However, Q-statistics suggests that the autocorrelation is present 
up to 24th lag. By implementing ARCH(6) model we were able to capture ARCH effect and 
volatility clustering of exchange rate time series. 
All parameters of variables (except for both constants) are highly statistically significant, and 
whole model is highly statistically significant as well. High value of R
2 indicates that fitted 
values of forward exchange rate almost totally fit the real values, as it can be seen from  
Fig. 5. However, there is one serious drawback of the selected model. Residuals clearly do not follow 
NID (0,1) distribution, namely because of an excessive kurtosis (and therefore JB test for 
normality is highly statistically significant). Since residuals’ distribution is not normal, the 
chosen model is not suitable for prediction. 
Fig. 5 Comparison of fitted and real values of forward exchange rate 
26
27
28
29
30
31
32
33
34
35
250 500 750 1000 1250 1500
FORWARD 3M Fit of FORWARD 3M
-.10
-.08
-.06
-.04
-.02
.00
.02
.04
.06
250 500 750 1000 1250 1500
FORWARD 3M - Fit of FORWARD 3M
Source: CNB, own calculations
 
4  Conclusion 
In this paper, we tried to find out, whether Covered Interest Rate Parity holds for CZK/EUR 
exchange rate. 
We have found that: (i) the simple comparison of exchange rates’ and interest rates’ ratios 
does not confirm the validity of CIRP, as it is given by equation (2), (ii) the comparison of 
predicted  spot  rate  and  real  spot  rate  indicates  that  foreign  exchange  markets  were  not 
effective, which violates important assumption of CIRP, i.e. possibility of substituting the Et+1 
by Ft, (iii) the OLS regression augmented by MA(1) process of residuals and by ARCH(6) 
model of residuals’ variance shows, that the interest rates’ differential and the spot exchange 
rate were dominant factors of determining the forward (known future) exchange rate. Thus, in 
this sense, the CIRP was valid in the Czech republic during selected period. 
However, the chosen model is not suitable for forward exchange rate prediction, since the 
distribution of residuals is not normal.  
The challenge for future research may be implementing more sophisticated model, such as 
Markow-switching model, to be able to make the task of doing some valid predictions of the 
forward exchange rate development less difficult. 
Literature 
1.  Arlt, J., Arltová, M.: Finanční časové řady. Praha: Grada Publishing, a.s., 2003, 220 p. 
ISBN 80-247-0330-0. 
2.  Arlt, J., Arltová, M.: Ekonomické časové řady. Praha: Grada Publishing, a.s., 2007,  
288 p. ISBN 978-80-247-1319-9. 
3.  Durčáková, J., Mandel, M.: Mezinárodní finance. 2. vydání.Praha: Management Press, 
2003. 396 p.. ISBN 80-7261-090-2. 4.  Frait, J.: Mezinárodní peněžní teorie. Ostrava: VŠB – TU Ostrava, 1996. 210 p. ISBN 
80-7078-395-8. 
5.  Komárková, Z.: Integrace finančního trhu České republiky s eurozónou [doktorská 
disertační práce]. Praha: VŠE, 2006. Dostupné z WWW: http://www.vse.cz/vksp. 
6.  Komárek, L. – Komárková, Z. (2008): Integrace finančního trhu vybraných nových 
členských zemí EU s eurozónou. Praha: Studie Národohospodářského ústavu Josefa 
Hlávky, 2008. (in print). 
7.  Stančík,  J.:  Determinants  of  Exchange-Rate  Volatility:  The  Case  of  the  New  EU 
Members. Czech Journal of Economics and Finance, 2007, vol. 57, no. 9-10, p.414-
433. ISSN 0015-1920. 
Školitel: 
Doc. Ing. Luboš Komárek, M.Sc., M.B.A., Ph.D. 

Covered Interest Rate Parity: The Case of the Czech Republic

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