Short Term Interest Rate and Market Price of Risk Evolution MSc. Student: Hirtan Mihai Alexandru...

Short Term Interest Rate and Market Price of Risk Evolution

MSc. Student: Hirtan Mihai AlexandruCoordinator: PhD. Professor Moisa Altar

The Academy of Economic StudiesThe Faculty of Finance, Insurance, Banking and Stock Exchange

Doctoral School of Finance and Banking

July 2010, Bucharest

-comparison of Central and Eastern European countries-

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

Literature Review

Equilibrium modelsVasicek (1977), Cox, Ingersoll & Ross (1985)

today’s term structure of IR is an output

they do not automatically fit today’s 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

No-Arbitrage modelsHo-Lee (1986), Hull-White (1990), Heath, Jarrow & Morton (1992) today’s term structure of IR is an input

designed to be consistent with today’s term structure of IR

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 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 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

Model and Methodology

),( ttfT is the derivative of with respect to T evaluated at T=t( , )f t t

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

),(

)],()([)(

tt

ttftdrEdtt T

MPR is defined:

),(),()],()(),([)( tdZttdttttttftdr T Mahdavi (2008)

Starting from Heath, Jarrow, Morton model (1992), Mahdavi found:

2 2 3( ) ( ) ( )5 71 2 3 4 6dr(t) [f (t,t) α α r(t) α r t ]dt α α r(t) α r t α r t dZ(t)T

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

)(),(),(

),(),(),(

),(),(),(

),(),(

),(

trttfttf

ttrttfttf

ttfttfttf

TtfTtfttf

T

T

T

tTT

Discretization:

2

1 2 3

2 3

4 5 6 7

( )( )

( )

( )

( ) ( )

r t rt dt

r t r r

t

t t

The MPR becomes:

21 2 3

2 35 74 6

( ) ( ) ( , ) [ ( ) ( ) ]

( ) ( ) ( ) ( )

( ) ( ) ( , ) ( ) [

Tr t r t f t t r t r t

r t r t r t t

r t r t f t t r t

21 2 3

2 35 74 6

21 2 3

( ) ( ) ]

( ) ( ) ( ) ( ) ,

( ) ( , ) [ ( ) ( ) ]

r t r t

r t r t r t t

r t f t t r t r t

2 3

5 74 6

21 2 3

( ) ( ) ( ) ( )

where (t) ~N(0,1), Letting 1, we define:

( 1) ( 1) ( , 1) ( ( ) ( ) )

r t r t r t t

t r t f t t r t r t

2 2 34 5 6 7

E[ ( 1)] 0

E[ ( 1) - ( ) ( ) ( ) ] 0

t

t r t r t r t

the moments conditions to implement GMM

71,..,

2 2 1 1 1x(t) ,r(t),r(t) ,r(t ),r(t )

the vector of of parameters

the vector of instrumental variables

Let:

x(t)) rαtrαr(t)α-α)t(υ

x(t))υ(tth( 3

72

6542 )(1(

1),

GMM uses the orthogonality condition to estimate the parameters

( ( , )) 0E h t

,),(T

1 W ' ),(

T

1)J(

1T

1

T

t

T

t

thth

nr. of orthogonality conditions, 10 > nr. parameters to be estimated, 7 the efficient estimates are obtained by minimizing the objective function

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”.

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

0%

4%

8%

12%

16%

20%

24%

ROBOR 1M WIBOR 1MBUBOR 1M PRIBOR 1M

Interbank Offer Rates Evolution: Jan 2003 - May 2010

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 0,006r(t+1)-r(t) RO -0,150 1,523 1,863 20,412 1162,606 0,000r(t+1)-f(t,t+1) 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 0,045r(t+1)-r(t) PL -0,036 0,229 -0,838 5,549 34,119 0,000r(t+1)-f(t,t+1) 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 0,055r(t+1)-r(t) HU -0,016 0,614 2,037 10,853 287,033 0,000r(t+1)-f(t,t+1) 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 0,017r(t+1)-r(t) CZ -0,018 0,166 -0,934 8,340 116,023 0,000r(t+1)-f(t,t+1) CZ -0,133 0,209 -1,745 7,279 110,548 0,000

Country

ADF PP KPSS

t-stat A t-stat BAdj. 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* 0.084638BUBOR -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 Table 5.3B - test equation includes intercept and trend

* significant at 10% level

** significant at 5% level

*** significant at 1% level

ρ1 ρ2 ρ3 ρ4 ρ5 ρ6

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

Stationarity tests

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

Romania & Czech Republic - the 7 param. model is correctly specifiedHungary 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 p-values indicate that the orthogonality conditions displayed are satisfied and the models are correctly defined.

Results

Romania Coefficient Std. Error t-Statistic Probα1 -0,0104 0,0067 -1,5518 0,1226α2 0,1808 0,1077 1,6784 0,0951 *α3 -0,6633 0,3883 -1,7080 0,0895 *α4 -0,0078 0,0032 -2,4699 0,0145 **α5 0,2470 0,0884 2,7948 0,0058 ***α6 -2,1261 0,6881 -3,0900 0,0023 ***α7 5,3642 1,6143 3,3230 0,0011 ***J-statistic 3,3214P-value 0,3447 Table 6.1

Czech Rep. Coefficient Std, Error t-Statistic Probα1 -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 1,7350P-value 0,6292 Table 6.4




Poland Coefficient Std. Error t-Statistic Probα1 -0,045566 0,005519 -8,25584 0 ***α2 1,690424 0,220536 7,665061 0 ***α3 -15,6734 2,147721 -7,297688 0 ***α6 0,005091 0,001094 4,654589 0 ***α7 -0,069152 0,017148 -4,032675 0,0001 ***J-statistic 3,560992P-value 0,61418 Table 6.5

Hungary Coefficient Std. Error t-Statistic Probα1 0,00864 0,00529 1,63355 0,10420α2 -0,19948 0,11624 -1,71607 0,08800 *α3 0,97565 0,61550 1,58514 0,11480α4 0,00003 0,00001 2,60791 0,00990 ***α6 -0,00804 0,00436 -1,84212 0,06720 *α7 0,05150 0,02888 1,78348 0,07630 *J-statistic 3,73762P-value 0,44268 Table 6.6




• 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

32 3642,51261,22470,00078,0)( rrrrRO

32 0742,638855,41181,00009,0)( rrrrCZ

32 069152,0005091,0)( rrrPL

32 069152,0005091,0)( rrrPL

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

• 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).

32

2

RO

)(3642,5)(1261,2)(2470,00078,0

)(66,0)(1808,0)(

trtrtr

trtrt

32

2

CZ

)(0742,63)(8855,4)(1181,00009,0

)(6689,14)(7706,00102,0)(

trtrtr

trtrt

32

2

PL

)(069152,0)(005091,0

)(6734,15)(6904,10455,0)(

trtr

trtrt

32HU

)(05150,0)(00804,000003,0

)(19948,0)(

trtr

trt

•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.

•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

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.

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.

Thank you !

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 EU’s 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

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, 457–473 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. Singleton, 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.

Short Term Interest Rate and Market Price of Risk Evolution MSc. Student: Hirtan Mihai Alexandru...

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