The Academy of Economic Studies
The Faculty of Finance, Insurance, Banking and Stock Exchange
Doctoral School of Finance and Banking
Short Term Interest Rate and Market Price
of Risk Evolution
-comparison of Central and Eastern European countries-
MSc. Student: Hirtan Mihai Alexandru
Coordinator: PhD. Professor Moisa Altar
July 2010, Bucharest
Objectives & Motivation
• empirical comparison on the behavior of the short term interest rates (IR)
on 4 Central and Eastern European countries: Romania, Hungary, Czech
Republic and Poland assuming the no-arbitrage condition
• We estimate the market price of interest rate risk (MPR) – the extra return
required for a unit amount of interest rate risk
• The interest rate is one of the key elements in every financial market
• Long maturity interest rates are the average future short term rates information about future path of the economy
• Interest rates are important for a correct assets valuation, understanding
of capital flows, financial decision making and risk management
• Because the interest rate is not traded we cannot eliminate its risk through
dynamic hedging - it will be useful to know how to price it
Page  2
Literature Review
Equilibrium models
No-Arbitrage models
Vasicek (1977), Cox, Ingersoll & Ross (1985) Ho-Lee (1986), Hull-White (1990), Heath, Jarrow
& Morton (1992)
 today’s term structure of IR is an output  today’s term structure of IR is an input
 they do not automatically fit today’s term  designed to be consistent with today’s
term structure of IR
structure of IR
 they are difficult to calibrate - due to
imprecise fit, errors may occur in
evaluating the underlying bonds with a
strong propagation on the options
pricing
 the drift of the short rate is not usually a
function of time
Page  3
 easy to calibrate
 drift of the short rate is , in general,
time dependent
• Chan et al. (CKLS1992) show that volatility of the IR is highly sensitive to the level
of r . Models with elasticity >1 capture the dynamics of the IR better than those
with values lower than the unit.
• Christiansen et. al (2005) indicates that the inclusion of a “volatility effect”
considerably reduces the level effect. Allowing for conditional heteroscedasticity in
the diffusion of the IR she found that the volatility elasticity is not significantly
different from 0,5 (in acc. with CIR (1985)).
• Duffee(1996) argues the power of the US Treasury Bonds to be considered as a
proxy for the short term rate. Contemporaneous correlations between yields on
short-maturity bills and other instruments yields have fallen drastically due to
market segmentation.
• Using a nonparametric approach Aid-Sahalia (1996) finds strong nonlinearity in
the drift function of the IR. Though, the drift has the mean reverting property leading to a globally stationary process
• Stanton(1997) shows that the monthly frequency considered does not have an
Page  4
adverse effect on the estimated parameters.
• Chapman et al. (1999) tested successfully the substitution of the short term rate
with 3 month and 1 month Treasury Bills, avoiding the microstructure problems.
• Ahn and Gao (1999) advanced a parametric quadratic drift model that captures
the performances of non-parametric one
• Ahmad & Willmot(2007) found that the market price of risk is not constant, varying
wildly from day to day and it is not always negative.
• Al-Zoubi (2009) indicates that the short term rate is non-linear trend stationary
and the introduction of a non-linear trend-stationary component in the drift function
significantly reduces the level effect in the diffusion model.
• Mahdavi (2008) analyzes the short-term rates in 7 industrialized countries and the
Euro zone using 1M LIBOR as a proxy for the short-term rate. His model is welldefined for all the positive values of IR and has a general structure, nesting many
of the previous short-term models. Also he determined that the MPR for each
country has a nonlinear structure in IR
Page  5
Model and Methodology
Starting from Heath, Jarrow, Morton model (1992), Mahdavi found:
dr ( t )  [ f T ( t , t )   ( t ) ( t , t )] dt   ( t , t ) dZ ( t ), Mahdavi (2008)
when arbitrage opportunities are ruled out, the expected change in the
riskless rate at time t is equal to the current slope of forward curve
(observable at time t) , minus a risk premium
f T ( t , t ) is the derivative of f ( t , t ) with respect to T evaluated at T=t
MPR is defined:  ( t ) dt  
Page  6
E [ dr ( t )  f T ( t , t )]
 (t , t )
dr(t)  [f (t,t)  α  α r(t)  α r (t ) 2 ]dt 
T
1
2
3
α  α r(t)  α r (t ) 2  α r (t ) 3 dZ(t)
5
7
4
6
Model Parametrization – Mahdavi (2008)
Restrictions
Vasicek (1977): dr = k (θ-r) dt + σ dZ
α3 = α5 = α6 = α7 = 0
Brennan - Schwartz (1979): dr = k (θ-r) dt + σrdZ
α3 = α4 = α7 = 0
Cox – Ingersoll – Ross (1985) dr = k (θ-r) dt +σ r 0.5dZ
α3 = α4 = α6 = α7 = 0
Chan et al.(1992); dr = k (θ-r) dt +σ r 1.5dZ
α3 = α4 = α5 = α6 = 0
Duffie – Kahn (1996) dr = k (θ-r) dt +(α+βr) 0.5dZ
α3 = α6 = α7 = 0
Ahn – Gao (1999) dr = k (θ-r) r dt + σ r 1.5dZ
α1 = α4 = α5 = α6 = 0
Discretization:
The MPR becomes:
 ( t ) dt  
 1   2 r ( t )   3 r (t )
2
f (t , t ) 
2
 4   5 r ( t )   6 r (t )   7 r (t )
T
3
f (t , T   )  f (t , T )

f (t , t )   f (t , t   )  f (t , t )
T
f (t , t )   f (t , t   )  r (t , t )
T
f (t , t )   f (t , t   )  r (t )
T
Page  7
T t
r (t   )  r (t )  f T (t , t ) 
 [ 1   2 r ( t )   3 r ( t ) 2 ]  
 4   5 r (t )   6 r (t ) 2   7 r (t ) 3  (t )


r ( t   )  r ( t )  f ( t , t   )  r ( t )  [ 1   2 r ( t )   3 r ( t ) 2 ]  
 4   5 r (t )   6 r (t ) 2   7 r (t ) 3  (t )

r (t   )  f (t , t   )
 ,
 [ 1   2 r ( t )   3 r ( t ) 2 ]  
 4   5 r ( t )   6 r ( t ) 2   7 r ( t ) 3  (t )


w here  (t) ~N (0,1), Letting   1, w e define :
 ( t  1)  r ( t  1)  f ( t , t  1)  ( 1   2 r ( t )   3 r ( t ) 2 )
E [  ( t  1)]  0
E [  ( t  1) -  4   5 r ( t )   6 r ( t )   7 r ( t ) ]  0
2
Page  8
2
3
the moments conditions to
implement GMM
Let:
   1 ,..,  7 
x(t) 

the vector of of parameters
1,r(t),r(t) ,r(t  1 ),r(t  1 )
2
2

the vector of instrumental variables
υ(t  1 )  x(t)


h(  , t )  

2
2
3
(υ
(
t

1
)
-α

α
r(t)

α
r
(
t
)

α
r
)

x(t)
4
5
6
7


GMM uses the orthogonality condition E ( h ( , t ))  0
to estimate the parameters
nr. of orthogonality conditions, 10 > nr. parameters to be estimated, 7
the efficient estimates are obtained by minimizing the objective function
1
J(  )  
T
Page  9
T

t 1
 '
1
h ( , t )   W T  

T
T

t 1

h ( , t )  ,

WT is a positive-definite symmetric weighting matrix
GMM options – Eviews:
• Newey-West procedure for finding a weighting matrix robust to
heteroskedasticity, serial correlation and autocorrelation of unknown
form (HAC)
•A prewhitening filter was used to run a preliminary VAR(1) prior to
estimation to soak up the correlation in the moment conditions.
•Quadratic spectral (QS) for a faster convergence and Newey&West ’s
fixed bandwidth.
•The iteration method was “sequentially updating”.
Page  10
Data
• One-month and two-month, monthly average national interbank rates:
ROBOR, WIBOR, BUBOR and PRIBOR covering Jan. 2003 – May 2010
• In the region the national bonds market has a poor development so we
can’t consider their rates as a benchmark for the IR nor for the MPR
• The forward rate was calculated using the 1-month and 2-month rates
assuming continuous compounding ƒ(t,t+1)=2∙r2M-r1M
• When 2M rate was not calculated through the fixing we used log-linear
interpolation between the 1M and the 3M rates: r2M=r1M1/2 ∙r3M1/2
Page  11
Interbank Offer Rates Evolution: Jan 2003 - May 2010
24%
20%
16%
12%
8%
4%
ROBOR 1M
BUBOR 1M
Page  12
WIBOR 1M
PRIBOR 1M
Ju
l0
9
Ja
n
10
Ju
l0
8
Ja
n
09
Ju
l0
7
Ja
n
08
Ju
l0
6
Ja
n
07
Ju
l0
5
Ja
n
06
Ju
l0
4
Ja
n
05
Ju
l0
3
Ja
n
04
Ja
n
03
0%
Table 5.1 - Summary statistics
Country
Page  13
Mean(%)
S.D.(%)
Skewness
Kurtosis
JB-test
Prob.
ROBOR 1M (%)
r(t+1)-r(t) RO
r(t+1)-f(t,t+1) RO
12,656
-0,150
-0,251
5,423
1,523
1,533
0,554
1,863
2,017
1,772
20,412
19,834
10,148
1162,606
1098,738
0,006
0,000
0,000
WIBOR 1M (%)
r(t+1)-r(t) PL
r(t+1)-f(t,t+1) PL
5,048
-0,036
-0,187
1,014
0,229
0,283
0,103
-0,838
-1,280
1,723
5,549
4,496
6,199
34,119
32,249
0,045
0,000
0,000
BUBOR 1M (%)
r(t+1)-r(t) HU
r(t+1)-f(t,t+1) HU
8,304
-0,016
-0,002
1,931
0,614
0,584
0,606
2,037
1,790
2,686
10,853
12,846
5,806
287,033
402,482
0,055
0,000
0,000
PRIBOR 1M (%)
r(t+1)-r(t) CZ
r(t+1)-f(t,t+1) CZ
2,437
-0,018
-0,133
0,726
0,166
0,209
0,741
-0,934
-1,745
2,796
8,340
7,279
8,200
116,023
110,548
0,017
0,000
0,000
Stationarity tests
ADF
Country
t-stat A
t-stat B
Adj. t-stat
A
ROBOR
-1.381923
-1.654476
-1.366319
WIBOR
BUBOR
-1.991070
-1.946240
-2.346756
-2.549730
PRIBOR
-1.182450
-0.996726
PP
Adj. t-stat
B
LM-stat A
LM-stat B
-1.641539
0.630101**
0.210595**
-1.882935
-1.619628
-2.113981
-1.894907
0.434902*
0.191741
0.084638
0.093045
-1.072624
-0.922685
0.197861
Table 5.3
0.146289*
A - test equation includes intercept
B - test equation includes intercept and trend
* significant at 10% level
** significant at 5% level
*** significant at 1% level
ROBOR
WIBOR
BUBOR
PRIBOR
ρ1
ρ2
ρ3
ρ4
ρ5
ρ6
0.944
0.886
0.826
0.773
0.718
0.667
0.947
0.869
0.776
0.676
0.577
0.477
0.932
0.838
0.749
0.649
0.547
0.443
0.956
0.894
0.823
0.744
0.663
0.578
autocorrelation coefficients until the 6-th lag
Page  14
KPSS
Even if IR have poor results on stationarity tests like ADF, PP, KPSS and
correlogram analysis – the problem is arguable:
• we are dealing with a finite discrete sample
• if the IR - a random walk with a positive drift it would converge to infinity
• if the IR - a driftless random walk then it allows for negative values
•The high results for the Jarque-Bera test for normality indicate that
almost all variables examined are not normally distributed. The only
exceptions for which the normality distribution hypothesis of the J-B test
can be accepted is Hungary (for 5% level of relevance). Though , the
kurtosis < 3 and skewness >0 indicate that the IR distribution is platykurtic
and skewed to the right.
•For all data sets the average short rate is lower than the lagged forward
one indicating a positive average risk premium for every interest rate
process
Page  15
Results
Romania & Czech Republic - the 7 param. model is correctly specified
Hungary and Poland - the 7 param. model could not explain the volatility
structure and we were forced to eliminate the irrelevant param.
checking the validity of our model
• taking T times (nr. of obs) the minimized value of the objective function
we get the Hansen test statistic . It states that under the null hypothesis
that the overidentifying restrictions are satisfied – T(number of
observation) times the minimized value of the objective function is
distributed χ2 with degrees of freedom equal to the number of moments
conditions less the number of estimated parameters.
The associated p-value expresses whether the null hypothesis is rejected
or not.
• The low values for the J-statistic of Hansen’s test and their associated pvalues indicate that the orthogonality conditions displayed are satisfied
and
Page
 16 the models are correctly defined.
Romania
α1
α2
α3
α4
α5
α6
α7
J-statistic
P-value
Table 6.1
Page  17
Coefficient
-0,0104
0,1808
-0,6633
-0,0078
0,2470
-2,1261
5,3642
3,3214
0,3447
Std. Error
0,0067
0,1077
0,3883
0,0032
0,0884
0,6881
1,6143
t-Statistic
-1,5518
1,6784
-1,7080
-2,4699
2,7948
-3,0900
3,3230
Prob
0,1226
0,0951
0,0895
0,0145
0,0058
0,0023
0,0011
*
*
**
***
***
***
Czech Rep. Coefficient Std, Error t-Statistic
Prob
α1
-0,0102
0,0024
-4,1833
0,0000 ***
α2
0,7706
0,1765
4,3651
0,0000 ***
α3
-14,6689
2,9867
-4,9115
0,0000 ***
α4
0,0009
0,0005
1,7632
0,0797 *
α5
-0,1181
0,0647
-1,8247
0,0699 *
α6
4,8855
2,6239
1,8620
0,0644 *
α7
-63,0742
33,5339
-1,8809
0,0617 *
J-statistic
1,7350
P-value
0,6292
Table 6.4
* significant at 10% level
** significant at 5% level
*** significant at 1% level
Page  18
Poland
α1
α2
α3
α6
α7
J-statistic
P-value
Table 6.5
Coefficient
-0,045566
1,690424
-15,6734
0,005091
-0,069152
3,560992
0,61418
Std. Error
0,005519
0,220536
2,147721
0,001094
0,017148
t-Statistic
-8,25584
7,665061
-7,297688
4,654589
-4,032675
Prob
0
0
0
0
0,0001
Hungary
α1
α2
α3
α4
α6
α7
J-statistic
P-value
Table 6.6
Coefficient Std. Error t-Statistic
Prob
0,00864
0,00529
1,63355
0,10420
-0,19948
0,11624
-1,71607
0,08800
0,97565
0,61550
1,58514
0,11480
0,00003
0,00001
2,60791
0,00990
-0,00804
0,00436
-1,84212
0,06720
0,05150
0,02888
1,78348
0,07630
3,73762
0,44268
***
***
***
***
***
* significant at 10% level
** significant at 5% level
*** significant at 1% level
*
***
*
*
• Similar to the results reported by Tse(1995), Nowman(1998), Kazemi,
Mahdavi, Salazar(2004) and Mahdavi(2008) we find that no single model can
explain the IR process in all Eastern European countries considered

RO
(r ) 
 CZ ( r ) 
 0 , 0078  0 , 2470  r  2 ,1261  r  5 , 3642  r
2
0 , 0009  0 ,1181  r  4 ,8855  r  63 , 0742  r
2
 PL ( r ) 
0 , 005091  r  0 , 069152  r

0 , 005091  r  0 , 069152  r
(r ) 
PL
2
2
3
3
3
3
• The volatilities functions for all the countries are nonlinear in the IR,
with high elasticity to its level but with different structures.
Page  19
• The drift of the IR for Romania, Czech Republic and Poland has a
quadratic structure in r. Though, the fact that the drift pulls back the short
term rate into the middle region when it goes for extreme values could
lead to globally stationary processes. This is according to the findings of
Ait-Sahalia(1996) and Ahn&Gao (1999)
• Hungary has the only direct mean reverting process due to linear drift in r
• We estimated the MPR of IR for each country defined as the extra
expected return required for a unit amount of interest rate risk
• The estimated lambdas are high nonlinear functions in the level of IR according to the results obtained by Kazemi, Mahdavi & Salazar (2004),
Ahmad & Willmot(2007) and Mahdavi (2008).
Page  20
 RO ( t )  
0 ,1808  r ( t )  0 , 66  r ( t )
2
 0 , 0078  0 , 2470  r ( t )  2 ,1261  r ( t )  5 ,3642  r ( t )
2
 CZ ( t )  
 0 , 0102  0 , 7706  r ( t )  14 , 6689  r ( t )
2
0 , 0009  0 ,1181  r ( t )  4 ,8855  r ( t )  63 , 0742  r ( t )
2
 PL ( t )  
 0 , 0455  1, 6904  r ( t )  15 , 6734  r ( t )
0 , 005091  r ( t )  0 , 069152  r ( t )
2
 HU ( t )  
2
3
 0 ,19948  r ( t )
0 , 00003  0 , 00804  r ( t )  0 , 05150  r ( t )
2
Page  21
3
3
3
Page  22
•Romania - The MPR is negative and relatively stable around the value of -0,4
suggesting a rational, risk averse behavior of investors. Negative peaks showing
the moments of fear appeared in delicate situations like the speculative attack
from September 2008 which had a strong impact across the entire region
•Poland, Hungary and Czech Republic - the situation is changing due to the fact
that MPR is positive revealing an aggressive behavior of the investors prepared
to take advantage on every occasion in these developing financial markets
•The MPR suffered a severe positive shock in 2004 in Poland&Hungary
immediately after they become EU full members in May 2004. This shock was
more severe in these countries due to the fact that they went for a cautious
capital account liberalization (a mandatory condition for EU adhesion) and they
were exposed to large speculative/investment inflows. The situation was not
replicated in Czech Republic who went for a rapidly liberalization of the capital
account in the early 90’s.
Page  23
•Czech Republic&Hungary: even though there are moments when the MPR is rising
and falling it seems that is returning to a middle range, showing a relative constant
attitude towards risk. The average lambda is 0,2 for Czech Republic and 13,5 for
Hungary, the last one being the largest one as an absolute value among the
analyzed countries.
• Poland: we can identify an attitude changing across the risk at the beginning of
2006, when average lambda is increasing from near 0 to 1,2 suggesting that
investors are willing to pay much more to take the risk
•The fact that investors are paying to take the risk reveals the hazardous behavior
described by Ahmad & Willmot(2007). We can mention anticipating interest rate
jumps or entering negative-expectation game pushed from behind by the
responsibility to their final clients. This does not turn out to be a winning bet all the
time because of possible interventions from the authorities or irrational behavior of
the market
Page  24
Conclusions
• We found evidence that no model can describe the short term interest rate
process in all the countries considered.
• More exactly even high-non linear volatilities with high elasticity with
respect to the interest rate level were found, they differ from case to case
as a structure.
• Estimating the MPR for each country, the results revealed a risk adverse
behavior of the investors in Romania in opposition to Poland, Hungary
and Czech Republic where “greedy” attitude was detected from the
investors.
Page  25
Limitation & Further research
• First of all we need to take a closer look about the periods/dates on which
the market price of risk had a high magnitude. Could structural changes of
short term interest rate cause them?
• We considered that the shocks on the interest rate are very frequent and
all the participants will adjust their expectations at least partially as an
answer to those shocks. Though by introducing dummy variables, besides
the risk to omit some of the shocks we faced difficulties in finding
economical motivation for all the structural changes
• Future research should consider an analysis that would relate the MPR
anomalies to the markets liquidity or to the lack of it. Also checking the
“level effect” using a GARCH model would be an interesting direction for
further analysis.
Page  26
Thank you !
Page  27
References
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