Econometric Modeling

Financial risk is currently at the center of all economic activity due to the incredibly unstable financial environment of the world economy. As a consequence the search for ways to reduce risk has taken a front seat in the important issues of our day. Several instruments exist in order to increase risk reduction possibilities, these include forward and futures contracts as well as various derivatives. That the most optimal number of risk reduction tools is used is vital. That ratio, the optimal number of risk reduction instruments, is decided by the relationship that exists between the spot instrument and the risk reduction tool. A time varying parameter model has been proven to be more effective in finding the relationship between economic variables and can therefore find an optimal reduction risk ratio that is not constant and can be controlled (Hatemi-J and Roca, 2006). The economic exposure can be controlled or regulated through hedging. In case of an existing futures contract in terms of a foreign currency, the entire hedging aspect turns out to be not so much of a concern. However, if no futures contract capable of being hedged is accessible, then the major issue which emerges is what futures contract to employ in order to hedge that specific currency’s risk (Ghosh, 1996).

The minimum variance hedge ratio of Johnson (1960) as well as the portfolio approach has been the ways that the stock index has been extensively studied in order to find risk reduction effectiveness. One way to lower risk is through the process of trading futures. The risk of price motion is lowered through this process. Initially, the stock index futures contracts were introduced as a way for those participating in the market to control their market risk without having to move the composition of their portfolios. Risk reduction, also known as hedging, only becomes truly important when there is a significant change in the value of a certain hedged item. The effectiveness of hedging is judged based on whether or not the hedging derivative offset the actual hedging. That hedging effectiveness as applied to futures contracts is the true way to determine financial futures contracts success was the argument poised by Pennings and Meulenberg (1997) (Kenourgios, Samitas and Drosos, 2005).

Simplified credit risk, high liquidity, as well as low cost is why stock index futures are one of the more successful financial derivatives used in the market. Even more so is the reason for futures offering tempting incentives to investors that allow them to reduce the risk exposure in the spot market, and also to allow them to hedge their portfolios. The way it works is through the compensation of favorable price movements in short future contracts sold compared to the unfavorable fluctuations in the longer units of stock index purchases. So the hedge ratio is the rough estimate comparing the amount of futures contracts that have been sold to the stock index. As for the actual influence of the hedge ratio on the reduction of risk depends on the technique adopted (Hull, 1997; Sutcliffe, 1997).

In giving a basic description, foreign exchange exposure can actually be defined as the level of sensitivity that exists for the real domestic value or worth of the currency of the assets, the liabilities and also the operating income with regards to the changes and fluctuations to the exchange rates that were not expected or anticipated. The risk that is related or linked with foreign exchange is measured by the dissimilarity or the variance that comes about from the domestic value or worth of the currency of the assets, the liabilities and also the operating income with regards to the changes and fluctuations to the exchange rates that were not expected or anticipated. A number of important facts that ought to be taken into consideration include the fact that the fluctuations or variations in the nominal exchange rate is not matched or balanced by the corresponding fluctuations or variations in the prices that are found domestically and also overseas as well (Adler and Dumas, 1984). More so, as will be discussed further in the paper, hedging whether it is operational hedging or financial hedging can result in a company having an increased value and worth bearing in mind the various and dissimilar market imperfections.

The key purpose of this project is an evaluation of whether or not econometric modeling of the hedge ratio produces any dissimilarity or variance with respect to money market effectiveness and cross hedgingof the exposure of Exchange rate risk. In the project, four different econometric models are used with the aim of estimating hedge ratio, namely, first difference model, error correction model, quadratic model and conventional model (levels). These models are derived, in the project, from the exchange rates and interest rates of three nations- Japan, Hong Kong and United Kingdom.

An error correction model is an econometric model that is kind of time series model with multiple variables in a direct form estimating the rate at which Y (the dependent variable) returns to its state of equilibrium when there are changes on X (the independent variable). This sort of model is valuable and advantageous for the short-term as well as long-term effects of a time series on another time series. Quadratic models can be defined as non-linear models, which take the elementary form when the functional element/component is non-linear and the parameters of the model are unknown. The conventional econometrics model used in the project is an OLS (ordinary least square) regression type, with the equation’s dependent variable being the unhedged exchange rate and the equation’s explanatory variable being the return rate or price of instrument which is being hedged. The instrument that is being hedged may be either futures contract or forward contract. Finally, the FD model takes into consideration the conventional model’s first differences.

The paper is categorized into 5 chapters. The first chapter introduces the topic of research, followed subsequently by a review of the literature in the second chapter. Chapter 3 comprises a discussion with regards to the empirical testing methods utilized in the study. Subsequently, chapter 4 contains the discussion and evaluation of the results acquired from empirical testing. The fifth and final chapter of this research project offers the summary and discussion along with key findings of this study.

Chapter 2: Literature Review

There are several categories for which the methods of empirically estimating of the hedge ratio that have been deduced. They fall into these following categories: 1. Error Correction (ECM) Models, 2. Autogressive Conditional Heteroskedasticity (ARCH) – based model, and 3. The Ordinary Least Squares (OLS) model. One of the main critiques for the OLS model is the fact that it does not consider the varying time distributions, heteroskedasticity, cointegration, and serial correlation. Thus, by not properly considering cointegration the model fails to specify correctly and results in a term known as underhedging. The ECM models have therefore been counted as superior since they tend to yield better results (Ghosh and Clayton, 1996; Chou, Dennis and Lee, 1996; Sim and Zurbruegg, 2001). The serial correlation is what the ARCH-based models focus on accounting for. Their varying distributions for time are therefore also better in yielding strong results than the OLS methods. Evidence does exist to support this argument (Baillie and Myers, 1991, Park and Switzer, 1995).

Theoretically it does seem that the results from the ARCH-based and ECM models should prove vastly superior to the OLS ones, the truth is that no method has actually been proven to be the best. An estimation of risk reduction ratios using the Greek stock and futures market was done by Floros and Vougas (2004). In their estimation they used ECM, OLS, BGARCH, and the VECM models. The results they found were that VECM as well as ECM seemed to give better results over the OLDS model. As far as the BGARCH model goes, it provided the best results out of all the different models. Another study was done by Lim (1996) in which the Nikkei 225 futures contracts were considered in regards to hedging performance. This study also yielded in favor of the ECM method. Finally, Rossi and Zucca (2002) argued the vast superiority of GARCH as a hedge ratio model over all other ones in the study they performed with the LIFFE traded futures contracts German Bund Futures, Eurolira, and the governmental bonds from Italy

Greece’s stock index futures market was used in Floros and Vougas’s study that examined its risk reduction ratios. This research study had the purpose of estimating either constant or time varying hedge ratios through various techniques. An array of econometric models was employed in order to estimate as well as derive the hedge ratios of both stock index futures contracts of the ADEX (Athens Derivatives Exchange). Vector and simple error correction models, multivariate generalized autogressive heteroscedasticity (M-GARCH), and regressions from standard OLS was used in the estimation of the proper corresponding hedge ratios usable. The M-GARCH models won out in superiority of its results for both of the stock index futures of Greece

An estimation of hedge performance was done based on the GARCH model by Lien, Tse, and Tsui (2002). Again, this work compares the hedge ratio performances from the constant-correlation VGARCH with the OLS model. VGARCH stands for vector generalized autogressive conditional heteroscedasticity. Out-of-sample hedge ratio forecasts were used to evaluate these methods. Currency futures, stock index futures, as well as commodity futures were used in the comparison of the two methods, and which also used the ten different spots in a systematic comparison. The results of the experimental study led to the knowledge that the assumption can’t be rejected in relation to eight out of ten series. The re-estimation of risk reduction ratios are thus kept current in order for the gain of maximum benefit of the varying time hedging strategy. Day by day rollover exists in which both OLS and VGARCH are re-estimated and the samples after re-estimation are studied. In this case it is found that the OLS hedge ratio has a better performance than the VGARCH equivalent. This result could be due to the great variety of the VGARCH models.

The Baltic Freight Index became Kuvassanos and Nomikos’s (2000) subject of study in their aspiration to compare the time varying and constant risk reduction ratios. They were to investigate the different ratios performance as far as reducing the freight rate risk went. The time-varying ratios used in this research study were based on a bivariate error correction model as well as a GARCH structure for errors. The two authors of the study also included a different GARCH version that placed errors into a conditional covariance matrix. This model was called (GARCH-X) Disequilibrium and uncertainty was linked together in this model in order to reflect the actual varying of time that exists in spot and futures pricing. The results for GARCH-X were promising and it showed that the model’s specifications did in fact provide even greater hedge reduction than a constant ratio like OLS or GARCH. One of the bigger weaknesses that became apparent though was the inability to eliminate riskiness of the spot position as much as other models have been recorded doing. Underlying indices with heterogeneous compositions might be to blame for this. So in order to amplify GARCH-X’s effectiveness in determining ratios, the composition of the Baltic Freight Index was restructured. The new structure reflected the shipping routes that were homogeneous, and may in fact increase the models effectiveness. This indicates what a beneficial impact on the market the introduction of the BPI (Baltic Panamax Index) has had on the Baltic International Financial Futures Exchange (BIFFEX).

Another study done by Chou, Denis, and Lee (1996) used the Nikkei Stock Average index from Japan to compare and contrast the error correction models. The NSA index was used as well as the NSA index with time intervals that varied. The results were overwhelming in favor of the error correction model over the conventional method. This then suggested that the hedge ratios that are achieved from the ECM model do in fact perform better than the conventional methods in procuring ratios. Temporal aggregation is also an important aspect to consider though and it has been found that it does affect the estimates of hedge ratios.

Another study using the ECM model in the international stock index was performed by Ghosh and Clayton (2004). These authors sought to apply the theory of cointegration in order to extend the price change risk reduction ratio. This method was applied to stock index futures contract for these following indices: NIKKEI, DAX, FTSE 100, and CAC 40. Short run deviations as well as equilibrium mistakes have been ignored in past research. This research study just like the ones aforementioned in this article, found that the ECM method was vastly superior to traditional models and conventional methods in procuring successful risk reduction rations. This was proven through the likelihood ratio test the researchers performed. Portfolios can thus be significantly more controlled by following the methods and models outlined in this paper, and they can do it all at a lower cost

OHR’s are time varying optimal hedge ratios that were estimate in Sultan and Hasan’s (2008) article. They used a version of the GARCH model which used an error correction term to save short-run differences in between a couple of linked assets. The GARCH error model was put up against other hedge ratio models that use the OHR constant, and they were tested in several different stock indices across Europe. Again the GARCH model seemed to have won by giving superior results in 3 out of the 4 different indices. The authors also compared and contrasted the GARCH-X model and found similar results as in the normal error correction GARCH model.

The English FTSE-100 stock index was used in an ex-ante hedging examination performed by Holmes (1995). The period of examination started in 1984 and ended in 1992. His results showed that it did not matter even if an ex-ante MVHR was used, if the portfolio that was to be hedged was underlying the contract futures, then there was a very large amount of risk reduction in comparison to a position that is unhedged. Holmes (1996) like many of the previous studies also attempted to appropriate the hedge ratios for the stock index through the GARCH (1,1) technique as well. The results were as following: MVHRs that used OLS in estimation outperforms risk reduction ratios that have been appropriated from other econometric techniques like the GARCH (1,1) error correction model. Additionally, the duration of the hedge impacted the effectiveness in a positive manner, but the correlation was still not straightforward (Kenourgios, Samitas and Drosos, 2005).

Holmes later cooperated with Butterworth (2001) in the creation of a similar article that focused on the effectiveness of hedging in the . This was done through the use of portfolios that had been greatly diversified as ITCs (). The LTSA approach was used in this scenario to appropriate risk reduction ratios and the results were promising. The contract used in this study provided a greater benefit to the stock index FTSE-100 when the portfolios reflected the FTIT as well as the Mid 250 (Kenourgios, Samitas and Drosos, 2005).

In an identical study, Butterworth and Holmes (2001) reviewed the hedging effectiveness involving the FTSE-Mid 250 stock index futures contract using Investment Trust Companies (ITCs) as their diversified portfolios. By making use of a substitute econometric technique (Least Trimmed Squares method) to approximate hedge ratios, their findings established that this contract is far better than the FTSE-100 index futures contact especially when hedging cash portfolios reflect the Mid 250, as well as, the (FTIT) indices (Kenourgios, Samitas and Drosos, 2005).

Cash portfolios that don’t meet any index in particular were first examined for hedging effectiveness by Graham and Jennings (1987). The portfolios that they created were unique in the way that the dividend payout and systematic risk differed from others. 90 equity portfolios of 10 stocks each were formed through random sampling. Following that they used the S&P 500 futures contract as a means to reduce the risk for each portfolio. MVHR was the model of choice when the ratios for hedging were calculated. The results provided were very interesting as they demonstrated in the way that the contracts weren’t even half as beneficial for portfolios that were real compared to those that had been hedged with cash indices. Additionally, Lindahl (1992) also appropriated hedge expiration and duration effects for the futures contracts of S&P 500 and MMI. The results were a direct correlation between hedge duration and effectiveness for the time period between 1 to 4 weeks. Although when it came to expiration it was not possible to find any visible patterns. These studies were further extended by Ghosh (1993) through the use of an ECM. His argument was that the OLS model was note effective because lagged values were ignored. This was demonstrated through the examination of the NYSE and the S&P 500 (Kenourgios, Samitas and Drosos, 2005).

The combinations of various capitalization portfolios that underlay five of the major indices were used in Figlewski’s (1984) appropriations on returns and risks. The period of examination lasted from June 1982 to September in 1983. OLS was the model used in order to procure the hedging ratios. The results of Figlewski’s work were that all indices ended up representing very diverse portfolios. Additionally, MVHRs showed more benefit than the ratios from the beta hedge model. Smaller stocks ended up having a larger risk attatched to them then the large capitalization portfolios. The dividend risk was almost a non-important factor whereas the maturity and duration of the hedge were of vital importance. The longer the hedges lasted, the more effective they ended up being. Furthermore, Figlewski (1985) constructed an additional study in which the duration of hedges was examined. The effectiveness of hedges that lasted between a day and as far as up to three weeks were examined. Again, the same result with the direct correlation of longer hedge duration equaled more effective risk reduction was discovered. Also the result that large scale portfolios had their risk reduced drastically compared to smaller portfolios was reaffirmed. Another article that reported using the OLS conventional model was put together by Junkus and Lee (1985) to investigate three stock indices in the U.S.A.’s effectiveness. Their results also supported the superiority of the MVHR mode, but they failed to discover a correlation between duration and effectiveness of the hedge (Kenourgios, Samitas and Drosos, 2005).

Later Bystrom (2003) was able to discover that the OLS ratios performed slightly better than the other models in his research study done considering electric futures contracts in Norway. The duration for his study lasted from 1996 to 1999. His work affirmed Holmes and Butterworth’s (2001) previous research data. They discovered that the OLS could result in hedge ratios that were significantly more reliable if the data outliers were excluded from the results. Additional information that supported the OLS claim of superiority was put forth by Lien, et. al (2002) who discovered that the model could even outperform the GARCH method.

An array of different specifications was used in Yang and Allen’s (2005) work that used the All Ordinaries Index to appropriate hedge ratios. Their specifications were as following: A vector autoregression, an OLS-based model, a vector error-correction model, and finally a diagonal-VGARCH model. They measured the effectiveness of the risk reduction through comparing risk-return and a maximization method for utility.

The ratios from the VGARCH model tends to outperform the constants in this case when it came to lowering the risk. However, if the return effects were to be considered equally then the utility-based OLS model seems to be favored inside of the sample hedge. Outside of those samples the VGARCH model wins in superiority both times

In order to create a proper and time varying hedge ratio without any non-linear restrictions, an ARCH model was used (Cecchetti et al. 1988). The information used in this type of application was taken from a treasury bond that was 20 years old in the United States of America. The duration lasted for one month. The estimations of the hedge ratio changes from 0.53 up to 0.91 over the duration, which demonstrates that the joint distributions of the spots and futures contracts are in fact time varying. However, the efficiency of maximizing hedge ratios through utility is here emphasized over the minimizing of variance that is available through the ARCH model. The previous text disputes the fact that minimizing variance is an efficient model in order to estimate ratios and thus claims that alternative models would be better choices.

Methodology

This chapter of the project is divided properly into sections which identify or categorize each of the phases of analysis and/or procedure of estimation. This consists of the methods that are used to measure whether the calculation or estimation of the hedge ratio using econometric models makes any difference or adjustment towards the effectiveness of the money market and cross currency hedging of exposure to foreign exchange risk. Tools such as Microsoft Excel as well as EViews will be employed in testing the econometric models. It is also important to point out that the variables that are used in this particular chapter are derivatives which are attained from the manipulation of the initial set of data that was retrieved from the databank in the Bloomberg website. They consist of exchange rates as well as rates of interests from the currencies of the three nations which are being analyzed and that is Canada, the United Kingdom and also Switzerland. In the subsequent chapter, there will be a detailed discussion on this data sent and the resulting manipulated set of data.

Hedge Ratio Estimation

In order to make an estimation of the hedge ratio in the money market, it is imperative and essential to calculate the forward rate with interest differential between x and y. The function between the two variables or parameters is attained as follows:

1

In the equation:

x denotes the Canadian dollar (CHD)

y denotes Swiss franc (CAF)

denotes the forward rate between the Canadian dollar and the Swiss Franc

denotes the spot rate between the Canadian dollar and the Swiss Franc

denotes the interest rate of Switzerland

denotes the interest rate of Canada

When hedging in the money market using the above information, would represent the exchange rate of the unhedged position whereas for the hedged position, the exchange rate would be denoted as .

In contrast, when hedging in the money market using the above information, also denoted as would represent the exchange rate of the unhedged position whereas for the hedged position, the exchange rate would be denoted as or .

Conventional Model (Levels)

The traditional Ordinary Least Squares regression model is estimated or calculated simply by attaining the regression of the logarithm of the spot returns and the logarithm of the future returns. This is signified by ?St. And ?Ft in a respective manner.

These are represented as follows:

St = St. — St.-1

Ft = Ft – Ft -1

The error term that is obtained from the regression model has a normal distribution with a mean of zero and a variance that is constant or homoscedastic so as to undertake hypothesis tests on the parameters of the econometric model. In addition, the error term is presumed not to have any correlation.

The conventional model takes the form of an Ordinary Least Squares regression model. The exchange rate of the unhedged position is represented by the dependent variable in the equation. On the other hand, the rate of return of the instrument being hedged or the price of the instrument is represented by the explanatory variable.

The following equations 2 and 3 represent the conventional model for the money market and for the cross currency in a respective manner.

2

3

In the first equation number 2, signifies the logarithm of the spot rate of the unhedged position or also denoted as . On the other hand, signifies which is the logarithm of the spot rate of the instrument being hedged where is the anticipated or estimated hedge ratio, in the manner that the coefficient of determination of the line of regression measures hedging effectiveness.

In the subsequent equation number 3, signifies the logarithm of the spot rate of the unhedged position or also denoted as. On the other hand, signifies which is mainly the logarithm of the spot rate of the instrument being hedged which is between the Canadian dollar and the British Pound where is the anticipated or estimated hedge ratio, in the manner that the coefficient of determination of the line of regression measures hedging effectiveness.

Conventional Model (First Differences)

In the previous model discussed above, the changing aspects in the short run are not taken into account. However, these changing aspects are in fact put into consideration in this econometric model being discussed. However, it is imperative to note that the first differences conventional model does not take into account changing aspects in the long run.

The following equations 6 and 7 represent the first differences conventional model for the money market and for the cross currency in a respective manner.

4

5

In the first equation number 4, signifies the logarithm of the spot rate of the unhedged position or also denoted as. On the other hand, signifies the first difference of which is the logarithm of the spot rate of the instrument being hedged where is the anticipated or estimated hedge ratio, in the manner that the coefficient of determination of the line of regression measures hedging effectiveness.

In the subsequent equation number 5, signifies the logarithm of the spot rate of the unhedged position or also denoted as. On the other hand, signifies which is the logarithm of the spot rate of the instrument being hedged which is between the Canadian dollar and the British Pound where is the anticipated or estimated hedge ratio, in the manner that the coefficient of determination of the line of regression measures hedging effectiveness.

Quadratic Model

This particular econometric model can be best defined as a model that takes the form where the term x is a variable when the other parameters c, b, and a are constants and where the parameter a is not equal to zero. With regards to estimating the parameters which are unknown, the least squares method is used.

The following equations 6 and 7 represent the qudratic model for the money market and for the cross currency in a respective manner.

6

7

In the first equation number 6, signifies the logarithm of the spot rate of the unhedged position or also denoted as . On the other hand, signifies which is the logarithm of the spot rate of the instrument being hedged where is the anticipated or estimated hedge ratio. The other term in the equation represents the hedging instrument’s quadratic value.

In the subsequent equation number 7, signifies the logarithm of the spot rate of the unhedged position or also denoted as and. On the other hand, signifies which is the logarithm of the spot rate of the instrument being hedged which is between the Canadian dollar and the British Pound where is the anticipated or estimated hedge ratio. The other term in the equation represents the hedging instrument’s quadratic value.

Error Correction Model

An error correction model is an econometric model that is kind of time series model with multiple variables in a direct form estimating the rate at which Y (which is the dependent variable) returns to its state of equilibrium when there are changes on X (which is the independent variable). This sort of model is valuable and advantageous for the short-term as well as long-term effects of a time series on another time series. This particular model is represented by the following equation:

Where:

Y represents the dependent variable

X represents the independent variable

EC represents the error correction component of the time series model and measures the rate at which the earlier derivations are corrected in order to attain equilibrium

The omission or exclusion of the error correction variable from any econometric model where the prices of instruments such as futures and spots are integrated together results in significant change of the estimated hedge ratios and the hedging performance as well. As a result, the estimation and calculation of a hedge ratio using the error correction model is centered on the two step method that includes long-run equilibrium relationship, changing aspects in the short run, non-stationarity and can offer an effectual estimate of the optimal hedge ratio comparative to the Ordinary Least Squares model. Therefore, it is important to test and consider whether the futures or spot employed in the research project are integrated and cointegrated prior to the specification of the model.

The following equations 8 and 9 represent the error correction model for the money market and for the cross currency in a respective manner.

8

9

In the first equation number 8, signifies the first difference of or which is the logarithm of the spot rate from a position that is not hedged that is there between, the spot rate between the Canadian and the Swiss currency. On the other hand, signifies the lag first difference of the spot rate. In addition, signifies the first difference of which is the logarithm of the spot rate of the hedging instrument where the element is the estimated or measured ratio of the hedge. On the other hand, signifies the lag first difference of the forward rate while denotes the error correction term.

In the subsequent equation number 9, signifies the first difference of or denoted as which is the logarithm of the spot rate from a position that is not hedged that is there between, the spot rate between the Canadian and the Swiss currency. On the other hand, signifies the lag first difference of the spot rate. In addition, signifies the first difference of which is the logarithm of the spot rate of the hedging instrument where the element is the estimated or measured ratio of the hedge. On the other hand, signifies the lag first difference of the hedging instrument while denotes the error correction term.

VaR Model

Different econometric models can be employed to not only calculate but also to estimate the hedging effectiveness. It is imperative to take note that when the rate of return or the minimum price variance of the hedged position is greater or higher than the rate of return or the minimum price variance of the unhedged position, then the hedge is considered to be more effective. This is realized if and only if the variance ratio satisfied the following condition:

Variance ratio > F (n-1, n-1)

The term in this case represents the size of the sample that is being considered.

The following equations 10 and 11 represent the variance ratio for the money market and for the cross currency in a respective manner.

10

11

In the initial equation number 10, the numerator signifies the variance of the rate of return of the unhedged position. The denominator, on the other hand, signifies the variance of the rate of return of the hedged position. In the subsequent equation number 11, similar to before, numerator signifies the variance of the rate of return of the unhedged position. The denominator, on the other hand, signifies the variance of the rate of return of the hedged position.

The hedging effectiveness of two dissimilar hedges can be undertaken on the basis of variance reduction. This can be calculated for both the hedged position and the unhedged position using the following equation:

12

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