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Short Term Interest Rate and Market Price of Risk Evolution

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### 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

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 models

Vasicek (1977), Cox, Ingersoll & Ross (1985)

No-Arbitrage models

Ho-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

- 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

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

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

Mahdavi (2008)

is the derivative of with respect to T evaluated at 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

MPR is defined:

The MPR becomes:

the vector of of parameters

the vector of instrumental variables

GMM uses the orthogonality condition

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

WT is a positive-definite symmetric weighting matrix

- 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

autocorrelation coefficients until the 6-th lag

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 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 p-values indicate that the orthogonality conditions displayed are satisfied and the models are correctly defined.

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.

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).

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.

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