RISK PREMIA IN THE CZECH MONEY MARKET

: We estimate risk premia in the Czech money market and we pay special attention to the 2008-2009 crisis period. Our results imply a rising forward premium and we argue that the error correction model is the most appropriate method, but median may be used as a first guess estimator. We estimated the term premium between the policy rate and various money market interest rates. In this context, ARCH models proved to be useful in reflection of non-stationarity observed in the data. The financial crisis caused a structural break in our data sample, but the impact on the forward premium was only brief and forward premia normalized quickly. The widening of the term premium proved to be much more persistent, although it declined significantly since the peak of the crisis.


Introduction
Money market interest rates serve as benchmarks for a significant portion of loans, bonds and derivatives (IMF, 2008).They are a key component of the monetary policy transmission channel.Disruptions, therefore, have a large impact on other segments of financial markets, the financial system, the economy and the effectiveness of monetary policy.Money markets were also the epicentre of the 2008/2009 financial crisis and have received a lot of attention since then.
There are several studies that deal with money markets in developed countries, but only few of them deal with the crisis in the context of emerging markets.
In our paper, we focus on Czech money markets and we pay special attention to the 2008/2009 crisis period.Our starting point is the expectation hypothesis of the term structure of interest rates.The expectation hypothesis offers an intuitive framework that forms the core of most standard interest rate structure models and is applied by practitioners in their investment decisions, although its empirical support is not unanimous in literature (Gravelle, 1998 or Kotlán, 1999).Using this framework, we estimate the forward premium (the difference between the expected market rate and corresponding forward rate agreement) and the term premium (the difference between the policy rate and key money market rates).Also, we test whether the premia changed during the crisis.

Biased expectation hypothesis
The starting point of our analysis is the biased expectation theory and we offer its brief overview in this section.There are more versions of expectation theory, but in the context of our paper the forward premium and term premium concepts are relevant (Costa, 2007).Biased expectation theory assumes that forward interest rates reflect the expected path of short-term interest rates and risk premia.In applying expectation theory restrictions, we may decompose the forward rate Standard expectation theory implies that a forward premium is higher for longer-dated instruments that bear more interest rate risks for the investor.
We assume a constant forward premium which is a standard approach.We try to derive market expectations from the actual development of interest rates after removing random noise, assuming a constant forward premium.Equation (1) may then be transformed into equation (2) for three-month interbank deposits represented by variable 3 h t y + , which builds the base of our empirical analysis.
A term premium is a premium based on the yield to maturity hypothesis that assumes that yield on a zero-coupon bond m t y (deposits) that matures in m periods (months) is equal to the average of one-period yields.In this context, term premium m t tp represents excess return over rolling over one-period bonds (deposits).We use linear approximation, but we agree that using compounded returns would be more precise.
We would expect both forward and term premium to increase with rising maturity.We assume them to be constant in our paper, but we discuss the impact of structural breaks.In our analysis, we compare the CNB's two-week repo rate and various deposit rates.The repo rate is the leading interest rate for market players and spreads below corresponding two-week deposits are pretty low, averaging 5-10bp over most of the sample period, although they widened significantly during the crisis period.

Methodology and data
We estimated the forward and term premium using different methods.We applied cointegration analysis, forward regression, ARCH model and basic descriptive statistics.In the Czech context, a similar approach was pplied by Kotlán (1999).
In contrast to him, we applied an error correction model that was employed by other researchers analysing foreign money markets.Early works applying cointegration technique similar to our approach may be found in Gravelle, Muller, Stréliski (1998) who deal with Canadian money markets.For Euro markets we found Costa, Galvão (2007) andDurré, Snorre, Pilegaard (2003) useful, although they deal with implied forwards.A comprehensive comparison of different instruments for US markets is provided by Gürkaynak, Sack, Swanson (2007).
Our data set includes three month PRIBOR and corresponding forward rate agreements (FRA) starting in one, three, six and nine months.Three month PRIBOR rates and other money market rates are from the CNB.FRA rates (ask) quotations and data used for illustrative purposes in the financial crisis section are from Bloomberg.We use monthly averages with a sample start in January 2000.There are three reasons for this choice.Firstly, the data in 1997 and 1998 were impacted by the 1997 financial crisis, when interest rates reached high double digit levels.Secondly, the Euro was introduced in 1999 and also the CNB had a one-year experience with inflation targeting.Both events could have caused structural breaks in the data sample.The sample ends in mid-2010, when the consequences of the last financial crisis continued to fade away.
The relatively high level of interest rates (see Figure 1) at the beginning of the sample was caused by the financial crisis in 1997 and the CNB's reaction to the crisis.

Forward premium
We decided to restrict our empirical tests on FRAs.In the absence of a futures market, FRAs are the most reliable reflection of market expectations.We avoided using implied forwards.Our experience shows that implied forwards are less stable than FRAs and results are model dependent.This is especially true for less liquid markets.

Descriptive statistics
The first useful way to look at the forward premium is to compare forward rates with actual interest rates in the corresponding period.Basic descriptive statistics are provided in Table 4.
The data support expectation of a rising time premium hypothesis.The spread between actual 3M PRIBOR and 1 to 4 months FRA is not significant, but grows strongly for more distant FRAs.The shape of the distribution changes depending on the forward's horizon.The distributions become flatter and skewed toward higher spreads for more distant forwards.This reflects lower accuracy of long-term expectations.Considering the shape of the distribution we think that the median is better estimator of the forward premium than the average.
The median is less sensitive to the positive skewness of the distribution and to occurrence of large observations.

Forward regression
The standard regression expectation theory test is based on Equation ( 2) that is transformed by subtracting the current level of interest rates (see Durre 2003) and rearranging terms.This transformation should overcome the possible non-stationarity of interest rates that could bias our estimates.
We estimate the equation on a horizon of 1, 3, 6 and 9 months corresponding to FRAs.The validity of the unbiased expectation theory would be confirmed by parameter values α=0 a β=1.A positive alfa parameter would imply a positive forward premium.The results are provided in Table 5 with parameter estimates in the first row and the t-test in the second.The third column includes R 2 statistics.
All results were obtained by using the Newey-West corrected covariance matrix to eliminate the bias caused by autocorrelated data.
The results support the expectations hypothesis.The beta parameter estimates are slightly above one and Wald's restriction test confirms that beta parameter estimates are not significantly different from one for all equations.We also imposed the restriction β=1 and estimated alfa parameters are in the last column (with reverse signs).They are equal to the sample averages and seem to have an upward bias as well.Models suffer from several weaknesses.Residuals are correlated and show ARCH effects.The Chow's and Quandt-Andrews breakpoint stability tests show that parameters are unstable.

ARCH models
We concluded in the previous section that forward premium estimates are probably biased and residuals show ARCH effects.Thus, we decided to include the ARCH effect explicitly to remove the bias.We added a standard conditional variance equation.
The ARCH effect was not significant for one month ahead FRA; for longer horizons we found only the first lag to be statistically significant.The results are shown in Table 6.The support for β=1 hypothesis is weaker than in the constant variance test, but it was not rejected for any model at a 5% significance level by the Wald test.By imposing the β=1 restriction, we obtained risk premium estimates that are significantly lower than in the previous case.

Cointegration analysis
Cointegration analysis has several advantages in comparison to the static approach.Firstly, it is suited for non-stationary data.Secondly, it enables to test causality and exogenity of variables.Thirdly, it interprets the relationship of variables in terms of long-term equilibrium and and short-term dynamic.
On the other hand, there are also important weaknesses.Cointegration tests lack robustness, results depend on sample choice and we are not able to effectively deal with non-linearity often observed in the data.
We test the expectations theory by using the error correction framework.The two-dimensional error correction model is described by the following set of equations.
Endogenous variables 3M Pribor and FRA interest rates appear on both sides of the equation.The error correction term is the first term on the right side with error correction parameters alfa.The rest of the terms track the correlation structure of variables.We choose the two-month lag length for all models to remove linear dependence.The lag length seems to be appropriate for all models with the exemption of 1 to 4 month forward, where the lag exclusion test clearly supported only a one month lag.This is no surprise because both instruments overlap for two months.
Our results are summarized in

Conclusion
All estimates are summarized in Table 8.The highest estimates of the forward premium were obtained by forward regression and mean statistics.We consider these estimates biased and unreliable after theoretical and technical assumptions were violated.We prefer the median as the first guess estimator, because the statistic is robust to asymmetry observed in the empirical distribution.ARCH models proved to be helpful in removing bias shown by forward regression, but ARCH models largely ignore short-term dynamic that is driven by the monetary policy cycle and correlated forecasting errors.Finally, we think that cointegration framework offers best results.Error correction models enable to separate longterm equilibrium and short-term dynamic.Our estimates are in line with the biased expectations hypothesis and we also confirmed that forward premia recovered relatively briskly from the financial crisis.On the other hand, all methods assume the forward premium to be constant which may not be true.Costa (2007) shows that the forward premium is time-varying and its size reflects probability of capital losses for bond holders.This probability naturally increases when interest rates are expected to go up and is usually accompanied by positively skewed distribution of future interest rate changes.The asymmetric probability of future changes is then reflected in positive skewness of distribution.
Relatively high estimates of the forward premium were also found for advanced markets (see Gürkaynak 2007).There are two possible explanations.Firstly, the forward premium also reflects the credit premium, but this is factor should be less important in the derivate market in comparison to the term premium.Secondly, investors may have overpredicted future interest rates in the past ten years that were marked by declining interest rates and unexpected disinflation over the period.

Term premium
The relation between policy rate and money market interest rates remains the key point of interest for market participants and policy makers.We define the term premium as the difference between the money market rate and policy rate or as the excess return of a term deposit above rolling-over two-week repo tenders.The key challenges for researchers are the impact of monetary policy, possible structural breaks, nonlinearities or heteroskedasticity in the data.We do not attempt to split the term premium into different categories because reliable risk-free benchmarks are not available for Czech money markets.
Firstly, we look at simple descriptive statistics.Secondly, we apply cointegration framework similarly to the forward premia section.Thirdly, we estimate ARCH models to reflect the nonlinearity observed in the data.

Descriptive statistics
The first step to estimate the term premium is to use simple descriptive statistics.
We subtract the average policy rate from the corresponding money market rate to derive spreads.Policy rate is represented by the two-week repo rate which is a less risky instrument in comparison to the interbank deposit.We use linear approximation with the following formal expression.
The results are summarized in Table 9.The spread of money market rates over policy rises with maturity.The distribution of spreads has high kurtosis and is skewed towards higher values.The mean may be biased upwards and so we prefer the median.The distribution gets flatter and volatility larger with rising maturity which reflects higher uncertainty of distant forwards.The weakness of simple statistics is that we assume stable distribution and that it may be biased by the presence of a relatively low number of tightening/easing periods in our sample.Sample statistics were surely impacted by the financial crisis and Table 10 confirms it.Pre-crisis distribution of term spreads is less skewed, has lower kurtosis and its mean is closer to the median.Both median and mean statistics lower than in the full sample estimates.

Cointegration
Theoretically, cointegration is well-suited for estimates of term spread in the money markets.Most importantly, it enables to decompose the relationship between policy rates and different money market rates on a short-term and long-term "equilibrium" part.On the other hand, we still assume the "equilibrium" premium to be constant.
The necessary condition for cointegration is non-stationarity of underlying variables.The tests are represented in the appendix and they support the non-stationarity hypothesis.Similarly to the forward premium section, we will use an error correction framework that may be described by the following set of equations.
The money market rate of maturity m and the average two-week repo rate appear on both sides of the equations.The equilibrium or cointegrating equation is described by with the error correction parameter alfa.The rest of the terms track the correlation structure of variables.
Cointegration tests failed to prove a cointegration relationship between policy rate and money market rates, although the necessary condition was met and fundamental logic also supports the expectation of a cointegration relationship.We think that the structural break during the crisis period is the key reason.We reduced the length of the sample to the January 2000 -June 2007 period.In this shorter sample, the data support cointegration hypothesis decisively.Results are in Table 11.
All models were estimated with a three-month lag that was supported by standard tests.The cointegration vector shows that term spread rises with maturity and is statistically significant.The last column shows p-values for parameter restriction.Data for all models support the hypothesis that the parameter β is not significantly different from unity which is in line with the expectation hypothesis.Only one-month maturity is an exemption.
The error correction parameters have expected sign, but only policy rate parameters are statistically significant.Thus, disequilibrium is eliminated by policy rates, which is consistent with expectation theory.Residuals seem to be uncorrelated, but suffer from ARCH effects and are non-normal (money market interest rate components).Although ARCH models proved to be a useful tool to capture some sources of non-stationarity in the data, they are not able to reflect the shift in the mean that occurred during the financial crisis.The structural break is clearly visible from the residual as shown in Figure 3.The shock was persistent, although it gradually faded away.The structural break may have impacted our estimates of the term premium and therefore, we estimated all models only for the pre-crisis period ending in June 2007 similarly to the cointegration analysis.The premia estimates were similar to full sample results, but ARCH effects became statistically insignificant.From this point of view, ARCH models seem to be a robust technique in case we deal with structural breaks and non-stationarity in the data.
A closer look at residuals also may give an idea of persistence of the structural shock.Model residuals reached the maximum mostly in October 2010, when the financial crisis also culminated in most segments of financial markets.
The elevated risk premia, as measured by residuals, started to gradually fade away.Normalization was faster for shorter maturities, which is also in line with evidence from money markets.Source: author's estimate

Summary
Table 14 provides a summary of our estimates.We may conclude that the term spread rises with maturity and that estimation methods differ in their sensitivity to structural breaks and non-stationarity in the sample.Full sample estimates that include the crisis period are higher for all methods and the mean is especially sensitive to extreme observations.The ARCH model provides similar estimates for the full sample and pre-crisis sample period, but the error correction model gives useful estimates only for the pre-crisis period.Again, we believe that the median may give a useful first guess estimate with other techniques giving more precise estimates, but we should keep in mind that every method has its weaknesses.Source: author's estimate

Concluding remarks
The last ten years were remarkable for Czech money markets.Key interest rates have normalized to levels typical in advanced economies.The financial crisis provided considerable stress for all segments of financial markets including money markets.Risk premia widened abruptly following the disintegration of markets after Lehman Brother's collapse.Unfortunately, we lack a reliable risk-free benchmark in the Czech money market and, therefore, it is impossible to provide an accurate decomposition of money market rates into a risk-free part and risk premium.Instead, we focused on key risk premia implied by the expectation hypothesis frameworkthe forward premium and the term premium.Our analysis showed that forward premia were relatively stable during the financial crisis and they normalized quickly.The results generally support the biased expectation hypothesis, but estimates differ significantly depending on estimation methods.We argue that medians may provide the first guess estimate of the forward premium, but ARCH models and cointegration techniques are better suited to deal with non-stationarity observed in the data.We consider the forward premium to be relatively high and we see two possible explanations.
Firstly, the forward premium also reflects the credit premium, but this factor should be less important in the derivative market in comparison to the term premium.Secondly, investors may have overpredicted the future interest rate in the past ten years, marked by declining interest rates and unexpected disinflation during the period.A similar conclusion can be drawn from the term premium analysis, although the impact of the financial crisis was heavier.Our estimates show a clear structural break during the crisis period, but elevated premia considerably declined later.Methods that explicitly or implicitly assume a stable term premium fail or provide biased estimates.Again, medians seem to be a relatively robust estimator that less sensitive to nonlinearities and parameter instability observed in the data.We found ARCH models to be useful as timedependent variance enables to absorb a significant part of the shock that occurred during the financial crisis.

Figure
Figure 3 Selected ARCH models residuals

Table 1 Descriptive statistics (part I) -money market interest rates in % Statistics cnb MM 1M MM 2M MM 3M MM 6M MM 9M MM 12M
The subsequent decline reflects remarkable disinflation and maturing of the Czech economy and financial markets, which pressed Czech interest rates

Table 2
Individual unit root tests (part I) -money market interest rates

Table 2
Individual unit root tests (part II) -FRAs more conclusive support for unit roots.We divided our sample into two subsamples.The first subsample includes money market interest rates and policy rate.The second subsample includes FRAs and three month PRIBOR.The results are summarized in Table2.All tests support the hypothesis that we deal with non-stationary data.On the other hand, we acknowledge that unit root tests are problematic from the methodological point of view and highly dependent on sample choice.

Table 3
Group unit root tests (part I) -money market interest rates

Table 3
Group unit root tests (part II) -FRAs Source: author's estimate

Table 4
Difference between FRAs and actual 3M Pribor Source: author's estimate

Table 5
Forward regression

Table 6
Forward regression with ARCH effect Abbreviations in lines denote FRAs of different maturity, i. e. FRA_1to4 -FRA on 3M PRIBOR starting in one month.Parameter estimates are in columns with tstatistics in italics.Estimates correspond to percentage points.
Source: author's estimate

Table 7 .
The table starts with Johansen's Granger causality from forwards to spot interest rates.This means that deviations from the equilibrium are corrected by spot rates.We also applied standard residual tests.The results of the Lagrange multiplier test do not indicate residual autocorrelation, but residuals suffer from ARCH effects and appear to be non-normal.
hypothesis that the parameter β is not significantly different from one which is in line with expectation hypothesis.Error correction term parameters imply that interest rates behave as expected.There is clear a Legend: Abbreviations in lines denote MM_3M for 3M PRIBOR and FRAs of different maturity (FRA_1to4 -FRA on 3M PRIBOR starting in one month…).*werejectH0 hypothesis denoted by the number cointegration vectors, parameters t-statistics are in italics.Estimates correspond to percentage points.Source: author's estimate Cointegration models are better suited to test the expectations hypothesis than simply regression, but parameter values may not be constant especially in the context of the last ten years.The financial crisis caused a clear structural break in the development of many financial variables including money market interest rates, including FRAs.The impact is clearly visible from the development of the cointegration vector shown in Figure2.Model residuals were exceptionally large as well, but only for equations that included money market interest rates.

Table 8
Forward premium estimates

Table 9
Descriptive statistics Source: author's estimate

Table 10
Term premium statisticspre-crisis sample

Table 12
Term premium estimates Source: author's estimate

Table 13
ARCH models -residuals in bp