Short Term Interest Rate and Market Price of Risk Evolution

1
Short Term Interest Rate and Market Price of Risk
Evolution
The Academy of Economic StudiesThe Faculty of
Finance, Insurance, Banking and Stock
Exchange Doctoral School of Finance and Banking
-comparison of Central and Eastern European
countries-

• MSc. Student Hirtan Mihai Alexandru
• Coordinator PhD. Professor Moisa Altar

July 2010, Bucharest
2
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

3
Literature Review

• Equilibrium models
• Vasicek (1977), Cox, Ingersoll Ross (1985)

• No-Arbitrage models
• Ho-Lee (1986), Hull-White (1990), Heath, Jarrow
  Morton (1992)

• todays term structure of IR is an input
• designed to be consistent with todays term
  structure of IR
• easy to calibrate
• drift of the short rate is , in general, time
  dependent

• todays term structure of IR is an output
• they do not automatically fit todays term
  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

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• Chan et al. (CKLS1992) show that volatility of
  the IR is highly sensitive to the level of r .
  Models with elasticity gt1 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 adverse effect on the
  estimated parameters.

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• 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 well-defined 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

6
Model and Methodology
Starting from Heath, Jarrow, Morton model (1992),
Mahdavi found
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
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Model Parametrization Mahdavi (2008) Restrictions
Vasicek (1977) dr k (?-r) dt s dZ a3 a5 a6 a7 0
Brennan - Schwartz (1979) dr k (?-r) dt srdZ a3 a4 a7 0
Cox Ingersoll Ross (1985) dr k (?-r) dt s r 0.5dZ a3 a4 a6 a7 0
Chan et al.(1992) dr k (?-r) dt s r 1.5dZ a3 a4 a5 a6 0
Duffie Kahn (1996) dr k (?-r) dt (aßr) 0.5dZ a3 a6 a7 0
Ahn Gao (1999) dr k (?-r) r dt s r 1.5dZ a1 a4 a5 a6 0
The MPR becomes
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the moments conditions to implement GMM
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• 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 NeweyWest s fixed bandwidth.
• The iteration method was sequentially updating.
  

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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 cant 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,t1)2r2M-r1M
• When 2M rate was not calculated through the
  fixing we used log-linear interpolation between
  the 1M and the 3M rates r2Mr1M1/2 r3M1/2

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Table 5.1 - Summary statistics Table 5.1 - Summary statistics
Country Mean() S.D.() Skewness Kurtosis JB-test Prob.

ROBOR 1M () 12,656 5,423 0,554 1,772 10,148 0,006
r(t1)-r(t) RO -0,150 1,523 1,863 20,412 1162,606 0,000
r(t1)-f(t,t1) RO -0,251 1,533 2,017 19,834 1098,738 0,000

WIBOR 1M () 5,048 1,014 0,103 1,723 6,199 0,045
r(t1)-r(t) PL -0,036 0,229 -0,838 5,549 34,119 0,000
r(t1)-f(t,t1) PL -0,187 0,283 -1,280 4,496 32,249 0,000


BUBOR 1M () 8,304 1,931 0,606 2,686 5,806 0,055
r(t1)-r(t) HU -0,016 0,614 2,037 10,853 287,033 0,000
r(t1)-f(t,t1) HU -0,002 0,584 1,790 12,846 402,482 0,000


PRIBOR 1M () 2,437 0,726 0,741 2,796 8,200 0,017
r(t1)-r(t) CZ -0,018 0,166 -0,934 8,340 116,023 0,000
r(t1)-f(t,t1) CZ -0,133 0,209 -1,745 7,279 110,548 0,000
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Stationarity tests
Country ADF ADF PP PP KPSS KPSS
Country t-stat A t-stat B Adj. t-stat A Adj. t-stat B LM-stat A LM-stat B
ROBOR -1.381923 -1.654476 -1.366319 -1.641539 0.630101 0.210595
WIBOR -1.991070 -2.346756 -1.882935 -2.113981 0.434902 0.084638
BUBOR -1.946240 -2.549730 -1.619628 -1.894907 0.191741 0.093045
PRIBOR -1.182450 -0.996726 -1.072624 -0.922685 0.197861 0.146289
A - test equation includes intercept A - test equation includes intercept A - test equation includes intercept Table 5.3
B - test equation includes intercept and trend B - test equation includes intercept and trend B - test equation includes intercept and trend B - test equation includes intercept and trend
significant at 10 level significant at 10 level
significant at 5 level significant at 5 level
significant at 1 level significant at 1 level
  ?1 ?2 ?3 ?4 ?5 ?6
ROBOR 0.944 0.886 0.826 0.773 0.718 0.667
WIBOR 0.947 0.869 0.776 0.676 0.577 0.477
BUBOR 0.932 0.838 0.749 0.649 0.547 0.443
PRIBOR 0.956 0.894 0.823 0.744 0.663 0.578
autocorrelation coefficients until the 6-th lag
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• 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 lt 3 and skewness gt0 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

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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 ?2 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 Hansens
  test and their associated p-values indicate that
  the orthogonality conditions displayed are
  satisfied and the models are correctly defined.

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Romania Coefficient Std. Error t-Statistic Prob
a1 -0,0104 0,0067 -1,5518 0,1226
a2 0,1808 0,1077 1,6784 0,0951
a3 -0,6633 0,3883 -1,7080 0,0895
a4 -0,0078 0,0032 -2,4699 0,0145
a5 0,2470 0,0884 2,7948 0,0058
a6 -2,1261 0,6881 -3,0900 0,0023
a7 5,3642 1,6143 3,3230 0,0011
J-statistic 3,3214
P-value 0,3447      
Table 6.1
significant at 10 level
significant at 5 level
significant at 1 level
Czech Rep. Coefficient Std, Error t-Statistic Prob
a1 -0,0102 0,0024 -4,1833 0,0000
a2 0,7706 0,1765 4,3651 0,0000
a3 -14,6689 2,9867 -4,9115 0,0000
a4 0,0009 0,0005 1,7632 0,0797
a5 -0,1181 0,0647 -1,8247 0,0699
a6 4,8855 2,6239 1,8620 0,0644
a7 -63,0742 33,5339 -1,8809 0,0617
J-statistic 1,7350
P-value 0,6292        
Table 6.4
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Poland Coefficient Std. Error t-Statistic Prob
a1 -0,045566 0,005519 -8,25584 0
a2 1,690424 0,220536 7,665061 0
a3 -15,6734 2,147721 -7,297688 0
a6 0,005091 0,001094 4,654589 0
a7 -0,069152 0,017148 -4,032675 0,0001
J-statistic 3,560992
P-value 0,61418      
Table 6.5
significant at 10 level
significant at 5 level
significant at 1 level
Hungary Coefficient Std. Error t-Statistic Prob
a1 0,00864 0,00529 1,63355 0,10420
a2 -0,19948 0,11624 -1,71607 0,08800
a3 0,97565 0,61550 1,58514 0,11480
a4 0,00003 0,00001 2,60791 0,00990
a6 -0,00804 0,00436 -1,84212 0,06720
a7 0,05150 0,02888 1,78348 0,07630
J-statistic 3,73762
P-value 0,44268      
Table 6.6
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• 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

• The volatilities functions for all the
  countries are nonlinear in the IR, with high
  elasticity to its level but with different
  structures.

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• 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 AhnGao (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).

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• 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 PolandHungary 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 90s.

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• Czech RepublicHungary 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

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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.

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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.

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Thank you !
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References Ahmad, R Wilmott, (2007) The
Market Price of Interest-rate Risk Measuring and
Modelling Fear and Greed in the Fixed-income
Markets, Wilmott magazine, 64-70, 2-6 Ahn, D.,
Gao, B. (1999). A parametric nonlinear model of
term structure dynamics. Review of Financial
Studies, 12, 721-762. Aït-Sahalia, Y. (1996).
Testing continuous-time models of the spot
interest rate. Review of Financial Studies, 9,
386-425. Al-Zoubi, H. A. (2009), Short term spot
rate models with nonparametric deterministic
drift, The quarterly review of economics and
finance, 49, 731-747 Andersen, T. G., Lund, J.
(1997). The short rate diffusion revisited,
Working Paper Northwestern University. Arvai,
Z., (2005), Capital account liberalization,
capital flow patterns, and policy responses in
the EUs new member states, IMF working paper,
European Department Chan, K. C., Karolyi, G. A.,
Longstaff, F. A., Saunders, A. B. (1992). An
empirical comparison of alternative models of
the short-term interest rate. Journal of
Finance, 47, 1209-1227. Chapman, D. A., Long, J.
B., Pearson, N. D. (1999). Using proxies for
the short-term rate when are three months like
an instant? Review of Financial Studies, 12,
763-806. Christiansen (2005). Level-ARCH Short
Rate Models with Regime Switching Bivariate
Modeling of US and European Short Rates. Finance
Research Group Working Papers F-2005-03,
University of Aarhus Cox, J. C., Ingersoll, J.
E., Jr., Ross, S. A. (1985). A theory of the
term structure of interest rates. Econometrica,
53, 385-408. Duffee, G. R. (1996). Idiosyncratic
variation of treasury bill yields. Journal of
Finance, 51(2), 527-551. Duffie, D., Kahn, R.
(1996). A yield factor model of interest rates.
Mathematical Finance, 6, 379-406. Hagan, P.,
West, G. (2006), Interpolation methods for curve
construction, Applied mathematical finance, vol.
13, no.2, 89-129
29
Hansen, L. (1982). Large sample properties of the
generalized method of moments estimators.
Econometrica, 50, 1029-1054.    Heath, D.,
Jarrow, R., Morton, A. (1992). Bond pricing and
the term structure of interest rates A new
methodology for contingent claims valuation.
Econometrica, 60,77-105    Hong, Y., Li, H.,
Zhao, F. (2004). Out-of-sample performance of
spot interest rate models. Journal of Business
and Economic Statistics, 22, 457473    Kazemi,
H., Mahdavi, M., Salazar, B. (2004). Estimates
of the short-term rate process in an
arbitrage-free framework. Forthcoming in
International Journal of Applied and Theoretical
Finance, 7(5), 577-589.   Koedijk, K. G., Nissen,
F. G. J., Schotman, P. C., Wolff, C. C. P.
(1997). The dynamics of short-term interest rate
volatility reconsidered. European Finance Review,
1, 105-130.   Liano, J. M. (2007), A Practical
implementation of the Heath-Jarrow-Morton
framework, proyecto fin de carrera, Universidad
Pontificia Comillas, Madrid, Espana     Nielsen,
H. B. (2007) Generalized Method of Moments
Estimation, Lecture Notes   Nowman, K. B. (1997).
Gaussian estimation of single-factor continuous
time models of the term structure of interest
rates. Journal of Finance, 52, 1695-1707.    Singl
eton, K. (2001). Estimation of affine asset
pricing models using the empirical characteristic
function. Journal of Econometrics, 102,
111-141.   Stanton, R. (1997). A nonparametric
model of term structure dynamics and the market
price of risk. Journal of Finance, 52,
1973-2002.   Tse, Y. K. (1995). Some
international evidence on the stochastic behavior
of interest rates. Journal of International Money
and Finance, 14(5), 721-738.   Vasicek, O. A.
(1977). An equilibrium characterization of the
term structure. Journal of Financial Economics,
5, 177-188.