Econometric Modeling in relation to Foreign Exchange Risk

Econometric Modeling and the Effectiveness of Hedging Exposure to Foreign Exchange Risk Does Model Specification Matter? For decades, the debates on unconditional and conditional econometric models for

estimating the optimal hedge ratio have been extensive. However, there is still no

consensus to date on the optimal hedge ratio model. This paper argues that optimal

hedge ratio is inconsequential to model specification – whether conditional or

unconditional, what is important is the correlation coefficient between the

underlying unhedged position and the hedging instrument. Four different models;

(i) Levels (ii) First Difference (iii) Non-Linear (iv) Error Correction Model, have

been employed in the estimation of hedge ratio and applied to two hedging

instruments; Money Market and Cross Currency Hedge. The empirical evidence

indicate that due to the strong correlation between the price of the unhedged

position with money market hedge, all four models exhibited similar hedge ratio

and hedging effectiveness. When correlation is weak as evident by cross currency,

the hedging effectiveness in all four models becomes inefficient. The discussion

highlighted in this paper is in reference to three major developed currencies: Swiss

Franc (CHF), British Pound ( GBP) and Hong Kong Dollar (HKD) with data

spanning from 2001 to 2009.

2015

Matthew Au De Jian RMIT University

5/28/2015

1 BAFI 2085: Research Project in Finance

Section 1: Introduction

The existence of foreign exchange risk management can be traced back to the

shortfall of Bretton Wood system and also the end of U.S dollar pegged to the gold. Foreign

exchange risk relates to the effect of unexpected exchange rate changes to the value of the

firm. Many multinational companies (MNCs) are strongly susceptible to foreign exchange

risk by virtue of their international operations. The management of foreign exchange risk for

these companies represents an integral part of their daily operations (Papaioannou, 2006).

Prudent management of these multinational companies (MNCs) requires currency hedging

for their foreign transactions to avoid potential adverse currency effect on their profitability,

market valuation and long term sustainability. The measurement of foreign exchange risk on

companies can be done through the use of Value at Risk (VaR) or Expected Shortfall (ES)

methods. According to Geczy et al (1997) multinational companies tend to hedge their

foreign exchange risk exposure through financial hedging such as forward and money market

rather than internal hedging strategies. Also, Wong & Broll (1999) documented other

multinational companies (MNCs) tend to hedge their foreign exchange risk exposure through

cross currency hedge techniques.

When MNCs ultimately decides to hedge against their foreign exchange risk

exposure, another important question typically arises. The question here remains as to what is

the hedging proportion and how the choice of model specification to derive the optimal hedge

ratio affects the hedging effectiveness. Many researches have surfaced over the decade on the

optimal hedge ratio derivation from a stock and futures contract perspective. Dale (1981) first

documented that hedge ratio should be derived from a price level model. He found that using

a hedge ratio derived from price level model leads a futures hedging effectiveness of

approximately 97 percent. Following his work, Hill and Schneeweis (1981) analysed the

same data set and found that the hedging effectiveness based on price levels model tend to be

overspecified due to the autocorrelation problem. They argued that the hedge ratio was

unreliable in the sense that it violates the Ordinary Least Square (OLS) assumption.

Although, the hedge ratio estimate may still be unbiased, it longer contains a minimum

variance. Therefore, they highlighted the importance of price change models when estimating

the optimal hedge ratio.


Broll et al (2001) then discussed the importance of non-linearity of financial time series. He

proposed that spot-futures exchange relationship are nonlinear, thus it is important to model

the effects of nonlinearity through the use of quadratic models when deriving the optimal

hedge ratio. Later on, Ghosh (1993) and Lien (1996) argued the importance of optimal hedge

ratio estimation through error correction model (ECM) when cointegration relationship

exists. They highlighted that suppose the spot and futures price are cointegrated, an errant

hedger who omits cointegration relationship will tend to adopt a smaller than optimal futures

positions ultimately resulting in a relatively poorer hedge performance. Thus, the use of ECM

is the true indispensable component when forming hedging strategies. The paper discussed up

to this point for model specification and hedge ratio are based on the view that the variance of

and covariance with are constant over time. Though, in reality they are time varying thus

must be modelled using dynamic statistical hedge model such as put forth by Kroner &

Sultan (1993), Park and Switzer (1995) and Chen et al (2014).

Despite of clear existence of evidence documenting the superiority of conditional

models over unconditional models, there is still no consensus no date. This is because there is

existing evidence such as Myers (1991) who suggests that transaction costs as a consequence

to portfolio rebalancing outweighs the benefit of dynamic hedging.

Here, rather than contributing to the endless debate of unconditional or conditional hedge

ratio models, this paper differentiates from the rest in the sense that its objective is to evaluate

the fundamental correlation coefficient between the unhedged position and hedging

instrument as discussed by Ederington (1979). He documented that the optimal hedge ratio

and hedging effectiveness is inconsequential to model specification but rather dependent the

relationship between the spot position and hedging instrument. Law and Thompson (2005)

also documented that when correlation coefficient between the price of spot position and

futures contract are weak, the risk reduction capabilities for various hedge ratio models tend

to be generally lower. Also, Ghosh & Clayton (1996) highlighted that fundamental

correlation between the price movement of spot instrument and futures contract is an

important factor to driving the hedging effectiveness. Third, Moosa (2003) whom analysed

similar proposition employed four different models in estimating the hedge ratio and found

that there was no significant difference in the hedging effectiveness produced by one model

compared to another. He commented:


“Although there are many theoretical arguments on which model specification suits

best for deriving the optimal hedge ratio, the topic is still elegant. Though, what determines

the success or failure of a hedge depends on the correlation coefficient between the price of

the unhedged position and the hedging instrument. (Moosa, 2003)”

Thus, the objective of this paper follows in spirit of Ghosh & Clayton (1996), Law &

Thompson (2005) and Moosa (2003) to ascertain whether model specification matters in

respect to deriving an optimal hedge ratio under the context of money market and cross

currency hedge. Four unconditional models are used to calculated the hedge ratio: (i) Levels

(ii) First Difference (iii) Quadratic and (iv) Error Correction Models (ECM) . Following that,

the hedging effectiveness will be measured by the hedging instrument to reduce the variance

on the underlying unhedged foreign exposure position through the use of variance ratio and

variance reduction. This paper entails the use of three developed currencies: Swiss Franc

, British Pound and Hong Kong Dollar with its data spanning from

1998 to 2009.

Following a series of robustness checks, the empirical results concluded that all four

models under money market hedge exhibit similar hedge ratios, hedging effectiveness and

variance reduction capabilities as a consequence to high correlation to the underlying

unhedged foreign currency exposure. The hedging effectiveness and variance reduction for

all four models under money market were approximately 99 percent. In contrast to cross

currency hedge, low correlation coefficient between the underlying unhedged foreign

currency exposure with the third currency led to all four models exhibiting different hedge

ratio, inefficient hedging effectiveness and variance reduction capabilities compared to

money market hedge. The results from an out-sample test with ex-ante hedge ratio illustrated

similar conclusion on the hedging effectiveness of ex-post money market and cross currency

hedge ratios.

The remainder of the paper proceeds as follows. In Section 2, the related literatures

are reviewed. Section 3 presents the basic concepts of financial hedging. The methodology or

research designs are discussed in Section 4. In Section 5, the data and empirical results on the

relationship between optimal hedge ratio and hedging effectiveness are presented. Section 6

provides the concluding remarks.


Section 2: Literature Review

For decades, there have been an extensive range of studies which have been

conducted on the calculation of optimal hedge ratio. Various approaches have been proposed

and used in many financial and non-financial market settings such as wheat, soybean and

even electricity. The optimal hedge ratio is defined as the quantity of the spot instrument and

hedging instrument which ensures that the total value of the hedged portfolio does not

change. The hedge ratio proposes the portion of hedging instrument in which minimizes the

variance of the underlying unhedged position.

2.1 Early Research Studies

The history behind hedge ratio can be traced back to Johnson (1960) who extended

the theory of hedging using Markowitz (1952) portfolio theory. Ederington (1979) who

examined the hedging performance of the New Futures Market (GNMA and T-Bills) then

further formalized Johnson „s theory and derived the minimum variance hedge ratio (MVHR)

which minimizes the variance of the spot portfolio. He explained that a minimum variance

hedge ratio can be defined as the ratio of the covariance between the spot and futures price to

the variance of the futures price. The objective of the hedge ratio is to determine the hedging

ability of the financial instrument to minimize the price risk associated with holding a pre-

determined spot portfolio.

The MVHR or can be shown as follows:

Where is the value of the hedged portfolio during period ; and are the log of the

spot and futures prices during period ; is the hedge ratio; and

corresponds to

the covariance between the log spot and futures price during period . The minimum variance

hedge ratio can be obtained by differentiating with respect to the hedge ratio and solving the

first order conditions, which can be written as

( )

( )

While solving the first order conditions lead to

[1]

[2]


Where, denotes the minimum variance hedge ratio. The MVHR can also be obtained from

an ordinary least square (OLS) regression where the spot and futures prices are the dependent

and independent variable respectively. The estimated slope coefficient is then multiplied by -

1 to obtain the hedge ratio. The negative hedge ratio reflects that when a long spot position is

taken, the opposite will be a short futures position. When this is done the coefficient of

determination, is an appropriate measure of hedging effectiveness ( . The measure for

hedging effectiveness can be defined as the percentage reduction of the variance on the

underlying unhedged position . Equation [4] shows the degree of hedging effectiveness:

Where is also the square of the correlation between the spot and futures price.

Thus, here it can be seen that the hedging effectiveness of futures contract is a function of the

relationship with the underlying unhedged spot position.

Subsequently to Ederington‟s theory, there have been numerous studies which followed suit

such as those developed by Dale (1981), Hill & Schneeweis (1981) and Witt et al (1987).

These studies first tried to analyse in particular the question of whether the hedge ratio should

be estimated from a price levels or price change model. For instance, Dale (1981) who first

studied the hedging effectiveness of three foreign currencies futures with hedge ratio derived

from a price level model. His results concluded that all three currency futures documented

significant hedging effectiveness of approximately 97 percent for both two week hedge and

four week hedge during the period of mid-1974 to mid-1980. Here, Dale‟s price level

regression can be expressed as follows:

Similarly to Ederington (1979), and denotes the spot and futures prices at time ;

represents the hedge ratio and signifies the residual term for the regression at time .

Correspondingly, Hill & Schneeweis (1981) who studied a common set of data with a price

changes regression found the hedging effectiveness to differ significantly to that of Dale

(1981) findings. H&S criticised that autocorrelation problem was evident when using price

levels model and that it violates the OLS regression assumption. They find that although the

[3]

[4]

[5]


hedge ratio estimated by Dale (1981) was still unbiased, it becomes an inefficient estimate

such that it does not contain a minimum variance. Also, they found that when estimating at

price levels, either price series often contains a unit root or non-stationarity. Hence, an errant

hedger who overlooks these issues will eventually be under hedged due to the upward bias on

the hedging effectiveness of price level model. H&S then suggest that the regression should

be estimated from a price change perspective such as:

Where and represents the spot and futures price changes at time ; represents the

hedge ratio and is the residual term for the regression.

Consequently to Dale (1981) and Hills & Schneeweis (1981) findings, Witt et al (1987) then

studied the theoretical and practical differences among the two frequently used specifications

to estimate a hedge ratio in the context of hedging agricultural commodities such as sorghum,

barley and cash price with corn futures price. Their findings concluded that the hedge ratio

derived from a price level perspective was as statistically significant as price changes in terms

of hedging effectiveness. They argued that the proper hedge ratio model estimation is a

function of the hedger‟s objective and the type of hedging instrument being used.

Following that, Broll et al (2001) provided some empirical evidence of nonlinear spot futures

exchange rate relationships. Their research were based on 6 major currencies over the period

of 1993 to 1999 found that five out of six currencies of developed countries do have spot-

future exchange rates relationship which are either convex or concave shaped. Moreover,

they believe more significant nonlinear spot futures exchange rates relationship would exist

for emerging market and transition economies currency because of illiquidity issues. Thus,

they suggest that hedge ratio should be derived from the following:

Where , , and are similar to that of Dale (1981) and Hill & Schneeweis (1981) while

represents the quadratic term on the futures prices to model the non-linear relationship

between and .

[6]

[7]


2.2 Evolvement on Early Research Studies

Later on, Ghosh (1993) who analysed several stock portfolios hedged to the S&P 500

Index Futures found that hedge ratio derived from traditional models as in the earlier pages to

be misspecified due to the ignorance of cointegration relationship. Such that, the short run

dynamics and long run relationship embodied within the error correction term are not taken

into account. The cointegration theory was first developed Engle & Granger (1987) who

illustrated that if two price series are integrated at the same order, there must exist an error

correction representation. According to Lien & Luo (1993), they favoured the use of ECM

when deriving the hedge ratio for spot stock and futures index hedging effectiveness due to

its clear relationship between spot and future prices. Lien (1996) then demonstrated that an

errant hedger who omits the cointgration relationship when using first difference model (eq

6) will result to a relatively poorer hedging performance compared to a hedger who takes into

account the cointegration relationship. Thus the hedge ratio regression under ECM should be

estimated as follows:

∑

∑

Where and represents the spot and futures price changes at time ; signifies the

hedge ratio derived from ECM; denotes speed of adjustment parameter from

disequilibrium; is the residual term on the ECM regression.

Other research findings that support the cointegration relationship for S&P 500 Index Futures

to that demonstrated by Ghosh (1993) include Wahab & Lasgari (1993) and Arshanapalli &

Doukas (1997). Also, Quan (1992) found cointegration relationship between the spot and

short term futures prices in crude oil market as well. Chou et al (1996) who studied numerous

Nikkei spot index portfolios with NSA index futures, agreed with the ECM model being

more superior over conventional models such as price changes. They documented that ECM

does a better job in reducing the risk associated with the underlying cash position by on

average 2 percent in contrast to price changes with data spanning from 1989 to 1993. Also,

Lim (1996) who studied similar Nikkei stock and futures data confirmed the superiority of

ECM method. Ghosh & Clayton (1996) who applied the cointegration theory in estimating

the hedge ratio using stock index futures for CAC 40, FTSE 100, DAX and Nikkei also found

that ECM hedging effectiveness to be superior over those estimated by conventional models.

[8]


Despite of the clear existence of evidence pointing to the superiority of hedge ratios

estimated with the use of ECM over those calculated from price levels and price changes,

many other researchers have criticized the assumption of constant variance of and covariance

between the spot and futures instrument when OLS regression is used. They highlighted that

homoscedasticity or non-constant variances are evident when using OLS regression to

estimate the price level, price change and error correction model (ECM). They underlined

that in reality, inherent structural changes or shocks in economic conditions are bound to

occur such that the hedge ratio changes over time upon receiving new information. Such that,

the hedge ratio is time-varying over time (Grammatikos & Saunders, 1983)(Brooks & Chong,

2001). Thus, more sophisticated alternative hedging models such as ARCH and GARCH

framework developed by Engle (1982) and Bollerslev (1986) should be used. While the

ARCH model received considerable attention as it models heteroscedasticity, GARCH model

were more frequently used since it permits more parsimonious description over ARCH

conditional variance equation with arbitrary linear declining lag structure as a result of Box

Jenkins ARMA terms.

In accordance to the use of GARCH, Kroner & Sultan (1993) compared the hedging

effectiveness of hedge ratio derived from a bivariate error correction model (ECM) fitted

with a GARCH error structure found to have a higher hedging effectiveness and variance

reduction compared to that of conventional models. Their research documented that within in

sample test, conditional hedge tend to outperform conventional OLS hedge by 2.5 percent.

While in an out-sample test, conditional hedge outperforms conventional OLS hedge by 1.5

percent. Their study was performed using five different currency spot and futures data over

the period of 1985 to 1990.

Similarly, Bailie & Myers (1991) applied the use of multivariate GARCH specification to

model the conditional the conditional covariance matrix for six commodities futures contract.

They illustrate the superiority of dynamic models over unconditional OLS models in terms of

hedging effectiveness. Additionally, Park & Switzer (1995) provided support for the

superiority of GARCH hedge ratio over OLS models in their study of hedging performance

using S&500 index futures and Toronto 35 index futures data. They commented that though

GARCH model is the most preferred, the potential utility gain from portfolio rebalancing

must outweight the losses arising from transaction cost.


2.3 Recent work in Hedge Ratio Estimation

Following the development of dynamic hedging models, recent researches have put

forth much more complex estimation methods some of which have yet to be proved of their

severe improvements. For example, Also, Lypny & Powalla (1998) examined the hedging

effectiveness of dynamic hedging strategy of an Error Correction Model fitted with a

GARCH (1,1) for German Index DAX futures found statistically and economically

superiority of the model over error correction model fitted with no GARCH and GARCH

fitted with no error correction term. They explain the adoption of ECM-GARCH delivers the

highest utility for both in and out sample periods even when transaction costs related to

rebalancing were included.

Also, Lien & Tse (1999) applied the price change, vector autoregrssive model (VAR), ECM

and ARFIMA-GARCH approaches using Nikkei Stock Average (NSA) index over the period

of 1989 to 1997 concluded that price change hedge ratio performed the worst as compared to

the other models. Couple with that, Floros & Vougas (2004) estimated the hedge ratio using

daily data on Greek stock and futures market from August 1999 to August 2001 over the debt

crisis period based on price change, ECM, Vector ECM (VECM) and multivariate GARCH

model (M-GARCH). They found to be most superior over other models.

Another paper who examined conditional hedge ratio modelling includes Laws &

Thompson (2005) who compared the hedge ratio obtained through OLS, GARCH, EGARCH

in mean and exponential weighted moving average (EWMA) models using FTSE 100 and

FTSE 250 stock index futures data. Their findings highlighted that EWMA to be superior

over others methods throughout the period from January 1995 to December 2001. Also,

Pradhan (2011) who focused on the impact of asymmetries on the hedging of S&P CNX

Nifty Index and its futures index using OLS, VAR, VECM and MGARCH. The outcome of

her research, based on 1871 daily observation spanning the period of June 2000 to April

2007, shows that asymmetric models such as MGARCH to provide the greatest portfolio risk

reduction and generates the highest portfolio return.


Other recent findings include Hou & Li (2013) who studied the hedging performance

of newly established CSI 300 stock index futures using wavelet analysis, price change, ECM,

constant conditional correlation (CCC), dynamic conditional correlation (DCC) and

BGARCH. Their empirical result concludes that short-run hedging horizon favours the use of

BGARCH while long run hedging horizon favours unconditional price change model. In the

same year, Kostika & Markellos (2013) who analysed the hedging performance of optimal

hedge ratio derived from an autoregressive conditional density (ARCD) which allows four

moments of conditional distribution of normalized error to be have higher hedging

effectiveness and variance reduction in contrast to GARCH, price changes and ECM models.

2.4 Inconclusive debates on Model Specification

To date, the debate on which method is really the best option is still ongoing. This is

because there are conflicting evidences which can be found from other literatures. For

instance, Kroner & Sultan (1991) applied the use of bivariate GARCH model for Japanese

Yen spot and futures return found it to be inferior to that of unconditional OLS based model

in term of risk reduction. In the same year, Myers (1991) who studied extensively on

Michigan‟s wheat commodity futures found only marginal improvement of GARCH model

in terms of hedging effectiveness over constant unconditional covariance hedge approach

estimated by OLS models. His findings concluded that GARCH hedging will not be

appropriate for risk adverse hedger due to the extra expenses arising from portfolio

rebalancing and complexity of using GARCH model.

Another prominent paper is from Holmes (1996) who analysed the ex-post hedging

effectiveness for UK FTSE 100 contract. He found that the risk reduction of a hedge strategy

based on hedge ratio estimated by unconditional OLS models outperforms advanced and

sophisticated techniques such as ECM and GARCH. Chakraborty & Barkoulas (1999) agrees

on the non-importance of utilizing sophisticated techniques such as GARCH (1,1) in

estimating hedge ratio as transaction cost associated with portfolio rebalancing will outweigh

the benefits. Their arguments were based on the empirical application to five leading

currencies spot and futures market data. The paper offered by Sim & Zurbrruegg (2001)

offers similar arguments on conditional and unconditional models on the FTSE-100 spot and

futures contract found that the latter shows significant advantage in hedging effectiveness

compared to the former.


Another paper by Lien, et al. (2002) explained that if conditional heteroscedasticity is a

characteristic of many financial time series, there is no clear superiority of conditional

models. They applied the use OLS, constant correlation model and GARCH in relevance to

ten spot and futures market covering currency, commodity and stock index futures and found

that the latter do not outperform the classical OLS and also constant correlation model. The

use of GARCH was further questioned as a result of expensive transaction cost due to

portfolio rebalancing. Other more recent papers which found no evidence that complex

econometric models have significant improvement over simple ordinary least square hedge

ratio includes Boystrom (2003), Alexander & Barbosa (2007), Harris, et al (2010), Chen, et al

(2014) and Wang, et al (2015).

Here, rather than contributing to the endless debate on conditional or unconditional

optimal hedge ratio model, this paper looks into the fundamental correlation coefficient

between the price of unhedged position with the price of the hedging instrument as discussed

by Ederington (1979). He highlighted that the optimal hedge ratio with hedging effectiveness

is a function of the correlation coefficient between the price of the spot and futures contract

eq [4]. Also, Ghosh & Clayton (1996) highlighted that the fundamental correlation between

the price movement of the spot instrument and futures contract is an important factor to

determine the hedging effectiveness. Moreover, Law & Thompson (2005) documented that

the reduction in risk for various hedge ratio models were generally lower as attributed to the

low correlation coefficient between the return on the investment portfolio and the hedging

indices. Additionally, .Moosa (2003) who analysed similar proposition employed four

different models found that hedge ratio and hedging effectiveness to be inconsequential to

model specification. He commented:

“Although there are many theoretical arguments on which model specification suits best for

deriving the optimal hedge ratio, the matter is still elegant. Though, what determines the

hedging success rate depends on the correlation coefficient between the unhedged and

hedging instrument (Moosa, 2003)”

Thus, the objective of this paper follows in spirit of Ghosh and Clayton (1996), Law

and Thompson (2005) and Moosa (2003) to ascertain whether hedge ratio is indeed

inconsequential to model specification using four constant unconditional models; (i) Levels

(ii) First Difference (iii) Quadratic and (iv) Error Correction Model (ECM).


The methodology entailed the estimation of the hedge ratio with the use of money market and

cross currency hedge as the hedging instrument instead of futures contract which have been

widely discussed by many researchers. Here, we will use Swiss Franc (CHF), British Pound

(GBP) and Hong Kong Dollar (HKD) to discuss the paper‟s objective. Prior to discussing the

methodology, it is important to understand the basic principles of financial hedging. Thus, the

next section highlights the concept of financial hedging techniques.

[Next page for Section 3 on the concepts of financial hedging]


Section 3: The Concepts of Financial Hedging

3.1 Forward Market Hedge

A forward contract is an agreement between two parties to exchange a specified

amount of a currency at a specified exchange rate on a specified date in the future. When a

corporation anticipate future need for or future receipt of a foreign currency, they can set up

forward contract to lock in the rate at which they can purchase or sell a particular foreign

currency.

An example from a receivables point of view, a corporation enters into a forward

hedge when it decides to insulated its foreign receivables from possible depreciation.

Thus, it will locks itself into a predetermine exchange rate known as forward rate at

which it can sell a specific foreign currency and exchange it to home currency ,

therefore allow it to hedge the foreign receivables due at time . By locking into a

forward contract, the uncertainty which lies within the future home currency value ultimately

changes to a certain home currency value since the forward rate is known a time . This can

be illustrated as follows:

(

)

(

) (

) [

( )]

Where

(

)

(

)

[9]

[10]


3.2 Futures Market Hedge

`Currency futures contracts are standardized contracts specifying a standard volume

of a specific currency to be exchange for another currency on a specific settlement date in the

future. Thus, currency futures contract are similar to forward contracts in terms of their

obligation, though differ from forward contracts in several ways. Firstly, currency futures

contracts are traded on an exchange, therefore are standardized. Forward contracts on the

other hand are private agreement between two parties, thus the agreement can be tailored to

individual needs. Since forward contracts are private agreements, there is a possibility a party

may default on its side of agreement. The default risk for futures contracts are close to zero

due to the existence of clearing house. Secondly, futures contract are marked to market

(MTM) hence settlements are on a daily basis until the end of the contract. In the case of

forward contract, settlement only occurs at the end of the contract. Third, futures markets are

more liquid than forward market. Therefore, futures hedger can close their position if their

contract timing fails to match the underlying exposure.

3.3 Money Market Hedge

Another alternative hedging technique is money market hedge. To hedge in the

money market, the corporation will have to borrow the present value of the foreign

receivables in the foreign country at time . The present value is calculated by

discounting the future value of the foreign receivables with the foreign interest rate (

applicable from to . Immediately after that, convert the borrowed present value of

foreign currency into home currency based on the current spot exchange rate between and

. Following that, invest the proceeds based on the home currency interest rate (

applicable from to . The borrowed foreign currency will be repaid with the proceeds

from the receivables ( paid by its foreign counterpart at . Here, the money market

hedge creates a foreign denominated liability (loan) to offset the foreign denominated asset

(receivables) (Eiteman, et al., 2013). A money market hedge involves the use of a contract

and a source of funds to fulfil the respective contract. In the above instance, the contract is a

loan agreement. The corporation seek the use of money market hedge to borrow in one

currency and exchange the proceeds to another currency. Funds to fulfil the contract – that is

to repay the loan are generated from the business operation or receivables.


The discussed can be written as:

(

)

(

)

Money market hedge can also be considered as a synthetic forward hedge or a

hedging technique of which mimics the characteristic of a forward hedge (Butler, 2012). This

is particularly true only when covered interest parity condition holds (Al-Loughani & Moosa,

2000). According to the theory of covered interest parity condition, the variation in exchange

rate between two currencies is mainly caused by the differential in the national interest rates

for securities of similar risk and maturity. Thus, arbitrage opportunities from interest rate

differential do not exist. As a consequence, the receivables will be the same for both

forward and money market when interest parity condition holds. As a result, the outcome

from equation [10] will be the same as equation [13]1 or as follows:

(

) (

)

3.4 Cross Currency Hedge

While Al-Loughani & Moosa (2000) proposed that money market hedging is an

effective hedging technique as forward hedging when covered interest parity (CIP) holds.

Chang & Wong (2003) highlighted that some currencies particularly less developed countries

(LDCs) may not be easily available due to its less matured or heavily controlled capital

markets. While this restricts the use of money market hedging, there are, however, alternative

options such as cross currency hedge where a third currency is introduced to act as a

hedge against the base and exposure currency.

The general idea behind this hedging technique is for any profit (loss) made on the

exposure to be offset by the loss (profit) made on the third currency position. For this

technique to work, the exposure and third currency must be highly correlated to the base

1 Appendix 1 illustrates an example of synthetic forward hedge when CIP holds

[12]

[13]

[14]

[11]


currency. The cross hedge can be in the form of forward, futures or options. According to

Eaker & Grant (1987) whom analysed the use of cross currency hedge between EMS

currencies, he presented that a third currency which belong to the European Monetary System

(EMS) will be an effective hedge to an exposure currency which also belong to the EMS.

Additionally, Moosa (2004) documented that the correlation of the third and exposure

currency to the base currency should be equal or more than 0.50 in order for the hedging

technique to be effective. Furthermore, Aggarwal & Demaskey (1997) concluded that

Japanese Yen to be an effective third currency to hedge against investment in Asian newly

industrialized countries (AIC) due it‟s to close economic integration. Brooks & Chong (2001)

also found that cross currency hedge between USD/DEM and USD/GBP are effective in

reducing portfolio risk due to the high correlation as a result of close economic relationship

between Germany and United Kingdom.

[Next page explains Section 4 methodology applied in this paper]


Section 4: Methodology

4.1 Optimal Hedge Ratio Estimation

The present study employs the regression discussed by Dale‟s (1981), H&S (1981),

Broll et al (2001) and Ghosh (1996) to estimate the optimal hedge ratio. Here, let and

represent the logarithmic prices of the unhedged position and the hedging instrument

respectively such that and denotes the rates of return on their prices. Thus, the

underlying regression for price levels, price changes and quadratic regression are written as

Where , and are the estimated hedge ratio while the represents the hedging

effectiveness or based on Ederington (1979) model. In the case of foreign currency

exposure, is the logarithmic spot exchange rate on the exposure currency expressed in

base currency or . On the other hand, the hedging instrument position will be

represented by money market and cross currency hedge instead of futures contract that have

been widely documented by many researchers. If money market hedge is used, such that the

offsetting position involves an interest parity forward rate ( ) consistent with covered

interest parity. Therefore

(

)

Where

(

) *

( )+

Where and represents the interbank interest rate for currency and .

[16]

[17]

[18]

[19]

[15]


If the hedging instrument is represented by a cross currency hedge, then a third currency

will be introduced. With the use of a third currency , a second exchange rate between

and or will be formed and used as a hedging instrument. Similarly, will be in

logarithmic form. Thus, can also be represented as:

(

)

An alternative method to estimating the hedge ratio for equation 16 and 17 is through

(Markowitz, 1952) portfolio theory. At first the representation to minimize the variance of the

portfolio value for equation [16] and [17] can be represented by:

Following that, the minimum hedge ratios can be derived by differentiating both equations

with respect to their hedge ratios and solve the first order conditions,

( )

( ) (

)

( )

( ) (

)

In which later will gives

As highlighted in the literature review, Ghosh (1993) and Lien (1996) documented

that an errant hedger who omits the cointegration relationship but uses price levels or price

changes model will result to a smaller than optimal position on the hedging instrument

ultimately leading to a poorer hedging performance compared to a hedger who takes into

account the cointegration relationship.

[20]

[21]

[22]

[23]

[24]

[25]

[26]


Thus, the hedge ratio from an ECM perspective should be estimated as follows:

∑

∑

( )

Where and represents the logarithmic rate of return on the unhedged foreign

exposure position and hedging instrument at time ; signifies the hedge ratio derived from

ECM; denotes speed of adjustment parameter as a function of disequilibrium between the

unhedged position and hedging instrument while is the residual term on the ECM

regression.

Lien (1996) then provided a theoretical analysis on equation [28] and suggests when the two

price variables adjust from disequilibrium then it can be written as follows:

Such that the ECM hedge ratio can be calculated as

(

)

(

⁄ )

Where represents the correlation coefficient between the residual terms and

.

[27]

[28]

[29]

[30]

[31]

[32]


4.2 Rate of Return Estimation

In the above, four unconditional hedging models have been discussed to derive the

optimal hedge ratio. Using the estimated hedge ratios, the rate of return on the unhedged and

the hedged position are then calculated as follows:

( (

))

Where represents the rate of return on the unhedged position, is the rate of return on

money market hedge and expressed as rate of return on cross currency hedge.

4.3 Variance Ratio (VR) and Variance Reduction (VD) Estimation

The null hypothesis focused in this paper will be to determine the hedging

effectiveness of money market and cross currency hedge on an underlying unhedged foreign

exchange exposure. To do these, the rate of return variance of the unhedged foreign exchange

exposure will be compared with the hedging instrument rate of return variance. This can be

formally written as follows:

H0: = [Null Hypothesis]

H1: > [Alternative Hypothesis]

Where, represents the rate of return on the unhedged position and is a function on the

rate of and . The represents the variance of the respective hedging instrument rate

of return. The null hypothesis will be rejected if:

Where VR is the variance ratio of the variance of , under no hedge case to the variance

obtained from a hedged case or while n is the sample size. If the ,

this indicates that the respective hedging instrument is effective in the sense that it reduces

the variation of the underlying foreign exposure position. The VR test can be complemented

with the variance reduction (VD) which is calculated as

*

+

[33]

[34]

[35]

[36]

[37]


Section 5: Data and Empirical Results

5.1 Data

The empirical analysis discussed in this paper is performed using monthly exchange

rates and interest rates sourced directly from Bloomberg. The data spans over the period of 29

May 1998 to 30 September 2009, a length of time approximately twelve years with a total of

137 monthly observations. There is no significance in either the choice of exchange rates or

sample period. The exchange rates employed for this study will be the Swiss Franc ,

British Pound and Hong Kong Dollar .

The exchange rates obtained were directly quoted in terms of per unit of U.S Dollar. For the

purpose of this study, the direct exchange rate for or must be calculated.

This can be done by dividing ⁄ with ⁄ . also represents the underlying

unhedged position. Other than that, the cross exchange rate for or must be

determined. Similarly, this involves dividing ⁄ with ⁄ . Here,

represents one of the hedging instruments. The second hedging instrument will be

represented by the interest parity forward rate between and using monthly

interbank interest rate belonging to those currencies, and .The interest rates are

deannualised and expressed in decimals. Following that, , and are then measured in

logarithmic form before it being used to estimate the hedge ratio via ordinary least square

(OLS) method under the four different models in eViews 8.

Table 1: The Variables – Definition and Specifications

Variables Definition and Specification

Exchange Rate for [Foreign Exposure]

Interest Parity Forward Rate for [ Hedging Instrument]

Exchange Rate for [ Hedging Instrument]

Switzerland one Month Interbank Interest Rate

United Kingdom one Month Interbank Interest Rate


Table 2: Descriptive statistics for , , and

iCHF iGBP

Mean 2.282 2.275 0.175 1.433 4.742

Median 2.310 2.304 0.168 1.188 4.805

Minimum 0.221 0.220 0.027 0.999 1.426

Maximum 1.895 1.863 -0.986 -1.021 1.970

Standard Deviation (SD) -1.306 -1.292 0.361 0.445 -1.028

Skewness 1.560 1.557 0.128 0.125 0.475

Kurtosis 2.679 2.670 0.231 3.410 7.563

No. of Observations 137 137 137 137 137

Figure 1: Switzerland and United Kingdom one month Interbank Interest Rate

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Figure 2: Graphical illustration for , and

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5.2 Preliminary Checks

5.2.1 Unit Root Tests

Table 3: ADF and KPSS Unit Root Tests

Augmented Dickey Fuller

(ADF)

Kwiatkowski Pihllips Schmidt Shin

(KPSS)

Variable t-statistics P value LM Statistics

Levels With Intercept

-0.137494 0.9420 0.782206

0.157667 0.9397 0.780912

-0.621351 0.8610 1.199675

First Difference

-13.08219 0.0000 0.233972

-13.12556 0.0000 0.233247

-12.14207 0.0000 0.131748

Critical Values

Test Models 1% 5% 10%

ADF -3.479 -2.882 -2.578

KPSS 0.739 0.463 0.347

Source: Appendix 2

Note: The null hypothesis for ADF test is by the presence of unit root while the null hypothesis for

KPSS test is by the presence of stationarity.

In order to avoid spurious regression results, it is necessary to first test whether the

variables are stationary or non-stationary. To test for stationarity, the Augmented Dickey

Fuller test proposed by Dickey Fuller (1979) and KPSS test proposed by Kwiatkowski et al

(1992) are used. The KPSS test is used as a complement to ADF test as the credibility of

ADF were questioned by Schwert (1987). The null hypothesis for ADF test is that the series

contains a unit root. While the null hypothesis for KPSS test is that the series is stationary.

Table 3 reports the unit tests on the logarithmic levels and first difference of , and 2.

The results indicate that all three variables are non-stationary at levels since ADF and LM-

statistics indicate insignificance of p value and LM statistics. To solve for non-stationary, the

2 The variables are in logarithmic form


variables are differentiated once. As expected, both ADF and KPSS test indicate that all

variables are stationary or an I(1) process.

5.2.2 Autocorrelation

Table 4: Autocorrelation Test using Durbin Watson and Breusch Godfrey LM Test

Variable DW test

(AR (1))

LM test Chi-Sq

Probability

Money Market Hedge

0.101 -

2.186 -

0.226 -

2.007 0.0263 (Lag 3)

Cross Currency Hedge

0.083 -

1.832 -

0.093 -

1.995 0.0134 (Lag 9)

Source: Appendix 3 and 4

Note: The DW statistic is always between 0 to 4. A value of 2 indicates no autocorrelation of order

one, AR (1), in the sample. The Breusch-Godfrey LM test null hypothesis represents no higher order

of autocorrelation.

For practical learning purpose, I have modelled for autocorrelation problems even

though it might not have substantial effects on the hedging effectiveness.. Based on Table 4

above, AR (1) was imminent in both levels and quadratic model for money market and cross

currency hedge as exhibited by their low Durbin Watson. As highlighted by Hill &

Schneweeis (1981) presence of autocorrelation will overstate the hedging effectiveness.

Moreover, although the hedge ratio estimate will still be unbiased, it longer contains a

minimum variance (no longer BLUE3). As a consequence, the variables required differencing

to solve for autocorrelation problem. After differencing the variables once, both first

3 Gauss Markow Best Linear Unbiased Estimate Theorem


difference and error correction model (ECM) show no evidence on any significant

autocorrelation of first order, AR (1), based its Durbin Watson that is close to 2.

Since the error correction model (ECM) for both money market and cross currency

hedge uses two lag dependent variables, I further tested for serial correlation beyond AR(1)

using the correlogram Q statistics and Breusch-Godfrey (1986) LM test. In reference to the

correlogram Q statistics for money market ECM, there appears to be a significant

autocorrelation at lag 3. To confirm this, I performed the Breusch-Godfrey LM test on the

lag, the result concluded that autocorrelation exist for lag 3. As a consequence, I reestimated

the model with an MA(3) term to correct the autocorrelation problem. For cross currency

ECM model, it was found to contain a significant serial correlation at lag 9 hence I

reestimated the model with an MA (9) term. After that, the LM test confirmed that there was

no significant autocorrelation at lag 9.

Table 5: Variance Reduction comparison for ECM models with and without MA terms

Type Variance Reduction (VD)

Money Market Hedge

ECM without MA(3) term 0.9999

ECM with MA(3) term 0.9999


ECM without MA(9) term 0.3838

ECM with MA(9) term 0.3868

Source: Appendix 5

Note: VD is calculated based on equation [37]

To check for any significant differences between the ECM models which does and

does not take into account of autocorrelation beyond lag one, I compare their variance

reductions (VD) for both money market and cross currency hedge. Table 5 documents the

results. As suggest by Moosa (2015), there is indeed not much difference between models

with and without MA terms. Both models were found to portray similar results. The

difference for cross currency was mainly attributed to the differences in hedge ratio estimate.

For the purpose of isolating the autoccorelation effects on the optimal hedge ratio coefficient

estimate, I will proceed with the ECM models with the autoregressive (AR) terms.


5.2.3 Normality Tests

Table 6: Jarque Bera Test

Type JB Skewness Kurtosis

Money Market Hedge

0.722 0.1444 3.208

169.342 -1.012 8.078

15.120 0.616 4.062

122.881 -0.621 7.523


71.132 -1.277 5.435

194.153 -1.337 8.206

54.888 -1.144 5.091

42.521 -0.794 5.255

Source: Appendix 6

Note: The null hypothesis for Jarque Bera test is by the presence of normal distribution in the errors.

The null hypothesis is rejected when JB-calc > 5.99.

Another important error testing to consider is normality test using the Jarque-Bera

(1980) test. The results shown in Table 6 for both money market and cross currency hedge

using four different models illustrated that the errors are not normally distributed. Except for

levels under money market hedge, others have non symmetrical error distribution While this

may affect the inference made on the hedge ratio, central limit theorem holds in this case

since the sample size is sufficiently large enough ( >50). Central limit theorem (CLT)

suggests that the coefficient to have approximately normal distribution even if the residuals

are not normal under the circumstances of large sample set (Stuart, 2014).


5.2.4 Cointegration Relationship

Table 7: Engle-Granger test for unit root in the residual of the cointegration equation

Type t statistics P-value

Money market: regression -12.745 0.000

Cross Currency: regression -6.065 0.000

Significance level 1% 5% 10%

Critical Values -3.479 -2.882 -2.578

Source: Appendix 7

Note: The null hypothesis of the test presented is by the presence of unit root in the residual of the

cointegrating equation using ADF test.

Cointegration relationship must be necessarily proven before estimating the hedge

ratio under error correction model (ECM) for both money market and cross currency hedge.

There are two commonly used way to test for cointegration. The Engle & Granger (1987)

methodology seeks to determine whether the residuals of the equilibrium relationship are

stationary, while Johansen (1988) methodology determines the rank of matrix consisting of

the cointegrating vectors in the error correction model.

In this paper, I will proceed with Engle & Granger‟s two step estimation

methodology. The first step is to ensure that both price series are difference stationary in the

same order. The second step then hinges on the results from the first step. The residual )

from the regression with the same order difference stationary of with and with are

then tested for stationarity at levels or process. The golden rule is that if the residual of

the regression ) is found to be stationary, than a cointegrating relationship exist.

Under money market hedge, the regression between and interest parity forward

rate, , was first tested for stationarity. As shown in Table 3, the unit root test concludes that

both variables are non-stationary at levels but stationary at first difference. Since both are

integrated of the same order, there might exist a linear combination between the two series.

To test for that, a regression was run between the first difference of and interest parity

forward rate, .


Following that, the residual from the regression ) was then tested for unit root.

Based on Table 7 above, the results concluded that the residual series from the regression was

found to be stationary thus a cointegrating relationship exists. The same inference was made

for the residual series between the first difference of and . Overall, error correction

models do exist for both money market and cross currency hedge.

[Next page for Section 5.3 Empirical Results]


5.3 Empirical Results

Money Market Hedge: Rate of Return Comparison

Figure 3: Rate of Return between Unhedged Position and Money Market Hedge based

on Levels Model


on First Differences Model

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Error Correction Model (ECM)



Cross Currency Market Hedging: Rate of Return Comparison

Figure 7: Rate of Return between Unhedged Position and Cross Currency Hedge based

on Levels Model


on First Differences Model

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Figure 10: Rate of Return between Unhedged Position and Cross Currency Hedge

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1/6

/19

99

1/1

2/1

99

9

1/6

/20

00

1/1

2/2

00

0

1/6

/20

01

1/1

2/2

00

1

1/6

/20

02

1/1

2/2

00

2

1/6

/20

03

1/1

2/2

00

3

1/6

/20

04

1/1

2/2

00

4

1/6

/20

05

1/1

2/2

00

5

1/6

/20

06

1/1

2/2

00

6

1/6

/20

07

1/1

2/2

00

7

1/6

/20

08

1/1

2/2

00

8

1/6

/20

09

Rat

e o

f R

etu

rn (

%)

Error Correction Model (ECM)



Figure 3 to Figure 10 illustrates a graphical comparison between the returns on the

unhedged position with returns from money market and cross currency hedge under four

different unconditional models. For money market hedge, Figure 3 to Figure 6 shows that the

return variances on the hedged positions are significantly reduced when using the hedge

ratios derived from the four different models. All four models depict a similar low volatility

over the sample period. Thus, this may suggest that money market is an effective hedging

instrument for the underlying unhedged foreign currency exposure.

In contrast to cross currency hedge, Figure 5, 6 and 8 illustrates that the return

variances on the hedged position still remains volatile as compared to the unhedged position.

Also, an interesting observation in Figure 7 shows that the variance of the returns of the

hedged position using hedge ratio derived from quadratic model has a higher foreign

exchange risk exposure compared to the unhedged position. Thus the higher degree of

hedging using the optimal hedge ratio derived from quadratic model, the higher the foreign

exchange risk exposure. Overall, cross currency hedge may not be an effective hedging

instrument to hedge the underlying unhedged position.

Table 8: The Estimated Hedge Ratios for H1 and H2

Hedging Method Estimated Hedge Ratio (h) t statistics

Money Market

Levels 1.0044 1519.210 0.999

First Differences 1.0003 1421.846 0.999

Quadratic 1.0372 175.629 0.999

Error Correction Model 1.0007 1372.820 0.999

Cross Currency Market

Levels 0.4699 11.054 0.475


Quadratic -2.4265 -2.531 0.509


Note: Esimated (h) is the coefficient derived from regressing the logarithmic return on the unhedged

position with the hedging instrument following the use of four models. measures the hedging

effectiveness between the unhedged and hedging instrument.


Table 9: The Estimated VD and VR for H1 and H2

Hedging Method Variance

(RU)

Variance

(RH) VD (%)

Money Market

Levels

7.397

0.001 11353.150 99.991


Quadratic 0.011 692.602 99.856


Cross Currency Market

Levels

7.397

4.559 1.622 38.365


Quadratic 85.608 0.086 -1057.369


Note: Estimated VR statistical significance is determined by comparing with f- distribution critical

value of 1. 327 (probability of 0.5 and d.f of 136).Whereas, VD measures the variance reduction

capability of the hedging instrument on the unhedged position.

Coupled with Figure 3 to 10, Table 8 and 9 depicts the descriptive results on the

hedge ratio , the hedging effectiveness ( ), the variances, variance ratio and

variance reduction for both money market and cross currency hedge using the four

different models Firstly, by looking at the variance of the unhedged position in contrast to the

hedged position regardless of money market or cross currency, it is no surprise that a no

hedge position has the highest volatility. Except for cross currency quadratic model, all

hedged position has a lower return variance as compared to a no hedge case. Thus the hedged

portfolios perform better than the unhedged position.

Secondly, the hedge ratios under money market hedge using four different models do not lead

to a vast dissimilarity and that the variance reductions are not all that different. In all cases,

hedging is effective and is approximately 99 percent throughout the sample period.

Also, all four models are statistically significant since their variance ratio far exceeds

the f distribution critical value of 1.327. Thus, the null hypothesis which stems that the

variance of the unhedged position is the same as the variance of the hedging instrument is

rejected. Also, the variance reductions appeared to be significant high or approximately

99 percent. Thus it is effective to hedge the unhedged position with money market hedging.

The strong hedging effectiveness and significant variance reduction (VD) here can be mainly

attributed to the strong correlation coefficient between the unhedged position and interest


parity forward rate between and . Here, the hedge ratios estimated under money

market are optimal.

Third, when the correlation coefficient between the unhedged position and hedging

instrument weakens, the hedging effectiveness ultimately declines. This can evident

based on cross currency hedge. Although, three out of four models under cross currency

hedge are statistically significant since VR exceeds the f-distribution critical value of 1.327.

The variance reductions (VD) for levels, first difference and error correction model (ECM)

are only approximately 39 percent throughout the sample period. This result is far from the

variance reduction obtained under money market hedge even though the null

hypothesis is rejected in this case. As a result, the hedge ratios under cross currency

hedge are suboptimal. Although the results here suggest that cross currency hedge may not be

suitable for the underlying unhedged position, it is not generalize across other cross currency

hedges. As highlighted by Eaker and Grant (1987), cross currency hedge may be effective

when a third and base currency are part of the same region such as the European Monetary

System (EMS).

Fourth, although the quadratic model under cross currency hedge remains as statistically

insignificant an interesting finding is that an increase in hedge position

actually leads to higher foreign exchange risk exposure. This argument is similar to what was

illustrated in Figure 9 which encompasses the comparison between the returns of unhedged

position with the return of the cross currency quadratic model.

Fifth, the results illustrated for money market hedge tend to mirror as what was documented

by Hill and Schneeweis (1981). As a consequence of serial correlation or autocorrelation of

order one, AR (1), the hedge ratio and hedging effectiveness of levels and quadratic model

tend to be higher than the first difference and error correction model (ECM). Thus, ultimately

an errant hedger whom ignores the serial correlation problem tend to be under hedged and

faces a maturity mismatch which requires him or her to regularly update their hedge position.

Overall, it can be concluded that the selection of models used to determine the

optimal hedge ratio are not that of significance and can be negligible here. Though, what

matters the most is the correlation coefficient ( ) between the price of the unhedged position

with the price of hedging instrument. High correlation invariably produces effective hedge


irrespective to how the hedge ratio is modelled while low correlation typically produces

ineffective hedge as evident by cross currency hedge.

5.4 Out of Sample Hedging Effectiveness

Figure 11: Outsample hedging effectiveness simulation

Table 10: Comparison of Out-of-Sample Hedging Effectiveness

Money Market Hedge Cross Currency Hedge

Variance

(RH)

Variance

Ratio (VR)

Variance

Reduction

(VD) %

Variance

Return (RH)

Variance

Ratio (VR)

Variance

Reduction

(VD) %

Levels 0.000 21103.942 99.995 6.302 1.522 34.281

First

Difference

0.000 22402.040 99.996 6.216 1.543 35.173

Quadratic 0.001 17912.153 99.994 62.053 0.155 -547.118

ECM 0.000 22576.245 99.996 6.226 1.540 35.068

Note: Estimated VR statistical significance are determined by comparing with f- distribution

critical value of 1.4944 (probability of 0.05 and d.f of 68).

Here, both the theoretical and empirical results on hedging effectiveness are explained

with an in-sample approach. In such an approach it is assumed that the optimal hedge ratio

and the hedging effectiveness can be determined in the same period. However, this is highly

unrealistic in practice and practitioner are concern is which method provides the greatest out-

of-sample hedging performance (Geppert, 1995)(Jong, et al., 1997).

May 29

1998

December

30, 2003

September

30, 2009

Sample for the

calculation of

the hedge ratio

68 months 68 months

Hedge


To conduct an out-sample hedging effectiveness test, the whole data set which spans

from 29 May 1998 to 30 September 2009 with 136 observations are separated into two

periods. The first period (29 May 1998 to December 30, 2003) is used as a specific period to

calculate the hedge ratio. After this period, the estimated hedge ratio is then applied onto the

underlying unhedged position over the second period (January 30, 2004 to September 30,

2009). In other words, the ex-ante hedge ratio calculated over the period of 68 months is then

applied over the remaining 68 months to simulate the hedge.

The ex-ante hedge ratio will be determined similarly to the ex-post hedge ratio with equation

[15], [16], [17] and [28]. The measure of hedging effectiveness will be based on the variance

of the returns on the unhedged position over the variance of the returns on the hedging

instrument given by equation [36] and equation [37].

Based on Table 10, the results concluded that all four models under money market are

effective in reducing the variance of the return on the underlying unhedged position by

approximately 99 percent throughout the second sample period (January 30, 2004 to

September 30, 2009). All four models‟ variance return are close to zero and are statistically

significant since their variance ratio far exceeds the f-distribution critical value of 1.494.

Similarly to in-sample hedging effectiveness, both first difference and error correction model

(ECM) have the highest variance reduction (VD) among the four models.

In contrast to money market hedge, the out-of-sample hedging effectiveness of cross currency

hedge for all four hedge ratio models is inefficient. The variance returns for all four models

are far from zero. Even though three out of four models are statistically significant, variance

reduction measured in percentage are only approximately 35 percent. The cross currency

hedge variance reduction ability is far from those obtained from money market hedge.

Correspondingly to ex-post hedging effectiveness results showed in Table 9, the ex-ante

hedging effectiveness here exhibit similar results for both money market and cross currency

hedge.


Section 6: Limitations

The initial focus of the paper was to include conditional ECM-GARCH models into

the discussion. However, money market ECM-GARCH was found to be statistically

insignificant while cross currency ECM-GARCH was found to have convergence issue

(Appendix 8). Thus, the recommendation here for future research is to include other forms of

conditional model for discussion even though optimal hedge ratio is inconsequential to model

specification.

Section 7: Conclusion

There have been a considerable number of studies on deriving the optimal hedge ratio

model, yet there is no consensus to date. Instead of contributing to the endless pit of

conditional and unconditional model specification debate, this paper revisits the issue by

investigating the fundamental correlation coefficient between the price of the unhedged

position and hedging instrument and then determines whether model specification does in

fact matter when deriving the optimal hedge ratio. Four unconditional minimum variance

hedge ratio models have been applied to two hedging instruments to determine its hedging

effectiveness in minimizing the variance of the unhedged position returns. Rather than futures

contract, the hedging instruments adopted here were represented by money market and cross

currency. The underlying unhedged position was represented by a foreign currency exposure.

The empirical results concluded that high correlation coefficient between the price of

the unhedged position with the interest parity forward rate ( ) or money market hedge

invariably produces effective hedges in all four unconditional models. When the correlation

coefficient weakens, the hedging effectiveness typically produces an inefficient hedge as

evident by cross currency hedge case. Furthermore, it have been found that ex ante hedge

ratio derived from all four models under money market hedge are as effective as ex-post

hedge ratio. First difference and error correction model (ECM) documented higher variance

reduction (VR) out of the four models. In contrast to ex-ante hedge ratio for cross currency

hedge, the variance reduction (VR) were bare minimal as compared to money market hedge.

Overall, the findings here is line with what was documented by Ghosh & Clayton, Law and

Thompson (2002) and Moosa (2003) who suggest that hedge ratio is inconsequential to

model specification but what is important is the correlation coefficient or the relationship

between the price of unhedged position with the price of the hedging instrument.


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Appendices


Appendix 1: Forward and Money Market Hedge Illustration

Swiss firm expects to receive £50,000 receivables from its British counterpart

CHF1.65/ £1

1.75 % GBP annual interest rate

2.50% CHF annual interest rate

Forward Market Hedging

⁄

⁄

⁄

Money Market Hedging

⁄


Appendix 2: Augmented Dickey Fuller (ADF) test

Null Hypothesis: S1 has a unit root

Exogenous: Constant

Lag Length: 0 (Automatic - based on SIC, maxlag=12)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -0.137494 0.9420

Test critical values: 1% level -3.478911

5% level -2.882748

10% level -2.578158

Null Hypothesis: F has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.882748

10% level -2.578158

Null Hypothesis: S2 has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.882748

10% level -2.578158


Null Hypothesis: D(S1) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.882910

10% level -2.578244

*MacKinnon (1996) one-sided p-values.

Null Hypothesis: D(F) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.882910

10% level -2.578244


Null Hypothesis: D(S2) has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.882910

10% level -2.578244



Kwiatkowski – Phillip – Schmidt – Shin (KWSS) test

Null Hypothesis: S1 is stationary

Exogenous: Constant

Bandwidth: 9 (Newey-West automatic) using Bartlett kernel

LM-Stat.

Kwiatkowski-Phillips-Schmidt-Shin test statistic 0.782206

Asymptotic critical values*: 1% level 0.739000

5% level 0.463000

10% level 0.347000

*Kwiatkowski-Phillips-Schmidt-Shin (1992, Table 1)

Null Hypothesis: S2 is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


Null Hypothesis: F is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000



Null Hypothesis: DS1 is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


Null Hypothesis: DS2 is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000


Null Hypothesis: DF is stationary

Exogenous: Constant


LM-Stat.



5% level 0.463000

10% level 0.347000



Appendix 3: eViews Regression Output

Money Market Hedge

(i) Levels

Dependent Variable: S1

Method: Least Squares

Date: 04/25/15 Time: 23:00

Sample: 1998M05 2009M09

Included observations: 137

Variable Coefficient Std. Error t-Statistic Prob.

C -0.000848 0.000545 -1.557770 0.1216

F 1.004404 0.000661 1519.210 0.0000

R-squared 0.999942 Mean dependent var 0.819730

Adjusted R-squared 0.999941 S.D. dependent var 0.105052

S.E. of regression 0.000806 Akaike info criterion -11.39352

Sum squared resid 8.78E-05 Schwarz criterion -11.35089

Log likelihood 782.4561 Hannan-Quinn criter. -11.37620

F-statistic 2307998. Durbin-Watson stat 0.100645

Prob(F-statistic) 0.000000

(ii) First Difference

Dependent Variable: DS1


Date: 04/25/15 Time: 23:05

Sample (adjusted): 1998M06 2009M09

Included observations: 136 after adjustments


C -3.30E-05 1.96E-05 -1.681662 0.0950

DF 1.000261 0.000703 1421.846 0.0000

R-squared 0.999934 Mean dependent var -0.002785








(iii) Quadratic



Date: 04/25/15 Time: 23:06

Sample: 1998M05 2009M09



C -0.012465 0.002141 -5.822877 0.0000

F 1.037165 0.005905 175.6289 0.0000

F2 -0.022331 0.004005 -5.576154 0.0000








(iv) Error Correction Model



Date: 05/24/15 Time: 14:06




C 2.08E-05 6.67E-05 0.311192 0.7562

DS1(-1) -0.103970 0.090345 -1.150809 0.2520

DS1(-2) -0.010251 0.089815 -0.114133 0.9093

DF 1.000691 0.000741 1350.906 0.0000

DF(-1) 0.105880 0.090426 1.170898 0.2438

DF(-2) 0.010868 0.089895 0.120893 0.9040

E(-1) -0.018205 0.022865 -0.796186 0.4274






F-statistic 335415.1 Durbin-Watson stat 2.006296




(i) Levels



Date: 04/25/15 Time: 23:09

Sample: 1998M05 2009M09



C 1.643820 0.074834 21.96618 0.0000

S2 0.469944 0.042512 11.05434 0.0000




Sum squared resid 0.787800 Schwarz criterion -2.248791







Date: 04/25/15 Time: 23:10




C -0.001255 0.001869 -0.671135 0.5033

DS2 0.580414 0.061570 9.426880 0.0000









(iii) Quadratic



Date: 04/25/15 Time: 23:11

Sample: 1998M05 2009M09



C -0.859893 0.831070 -1.034682 0.3027

S2 -2.426546 0.958653 -2.531204 0.0125

S22 -0.831183 0.274842 -3.024223 0.0030











Date: 05/24/15 Time: 13:06




C -0.000247 0.003631 -0.067929 0.9459

DS1(-1) 0.065799 0.086395 0.761611 0.4477

DS1(-2) 0.222903 0.082525 2.701020 0.0079

DS2 0.551906 0.058373 9.454852 0.0000

DS2(-1) -0.237324 0.075637 -3.137659 0.0021

DS2(-2) -0.133976 0.077606 -1.726356 0.0867

E2(-1) -169.6698 380.6605 -0.445725 0.6566









Appendix 4: Breusch-Godfrey LM Test

Money Market ECM

Before adding MA(3) term

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 3.060139 Prob. F(3,124) 0.0308

Obs*R-squared 9.236912 Prob. Chi-Square(3) 0.0263


After adding MA(3) term





Cross Currency Hedge ECM

Before MA (9) term





After adding MA (9) term





Appendix 5: Money Market and Cross Currency ECM



Date: 05/24/15 Time: 13:00



Convergence achieved after 9 iterations

MA Backcast: 1998M05 1998M07


C 4.11E-05 7.95E-05 0.516665 0.6063

DS1(-1) -0.096056 0.091898 -1.045236 0.2979

DS1(-2) -0.005874 0.092328 -0.063618 0.9494

DF 1.000664 0.000729 1372.820 0.0000

DF(-1) 0.097986 0.091974 1.065364 0.2887

DF(-2) 0.006081 0.092431 0.065792 0.9476

E(-1) -0.025380 0.027341 -0.928278 0.3550

MA(3) 0.219810 0.092347 2.380271 0.0188










Date: 05/24/15 Time: 13:33




MA Backcast: 1997M11 1998M07


C 0.002979 0.004363 0.682806 0.4960

DS1(-1) -0.000833 0.084713 -0.009834 0.9922

DS1(-2) 0.251557 0.079397 3.168327 0.0019

DS2 0.484070 0.053898 8.981247 0.0000

DS2(-1) -0.214646 0.068171 -3.148653 0.0020

DS2(-2) -0.217685 0.070080 -3.106226 0.0023

E2(-1) -583.8781 443.3917 -1.316845 0.1903

MA(9) 0.455770 0.091644 4.973259 0.0000









Appendix 6: Jarque Bera Normality Test

Money Market Hedge

(i) Levels


0

4

8

12

16

20

24

-0.001 0.000 0.001 0.002

Series: Residuals

Sample 1998M05 2009M09

Observations 137

Mean -7.98e-17

Median 9.94e-05

Maximum 0.002463

Minimum -0.001698

Std. Dev. 0.000803

Skewness 0.144080

Kurtosis 3.208605

Jarque-Bera 0.722404

Probability 0.696838

0

5

10

15

20

25

30

-0.0010 -0.0005 0.0000 0.0005

Series: Residuals

Sample 1998M06 2009M09

Observations 136

Mean 1.12e-18

Median 7.33e-06

Maximum 0.000760

Minimum -0.001032

Std. Dev. 0.000227

Skewness -1.012003

Kurtosis 8.078130




(iii) Quadratic


0

4

8

12

16

20

-0.001 0.000 0.001 0.002

Series: Residuals

Sample 1998M05 2009M09

Observations 137

Mean -1.13e-16

Median -6.25e-05

Maximum 0.002203

Minimum -0.001612

Std. Dev. 0.000724

Skewness 0.616351

Kurtosis 4.062685



0

4

8

12

16

20

24

-0.0005 0.0000 0.0005

Series: Residuals

Sample 1998M08 2009M09

Observations 134

Mean -1.42e-06

Median 7.33e-06

Maximum 0.000752

Minimum -0.000920

Std. Dev. 0.000215

Skewness -0.595014

Kurtosis 7.277815





(i) Levels


0

2

4

6

8

10

12

14

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Series: Residuals

Sample 1998M05 2009M09

Observations 137

Mean 6.40e-17

Median 0.012122

Maximum 0.128882

Minimum -0.268331

Std. Dev. 0.076109

Skewness -1.277650

Kurtosis 5.435478



0

4

8

12

16

20

24

-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

Series: Residuals

Sample 1998M06 2009M09

Observations 136

Mean 9.18e-19

Median 0.000665

Maximum 0.063802

Minimum -0.104426

Std. Dev. 0.021638

Skewness -1.337753

Kurtosis 8.206153




(iii) Quadratic


0

2

4

6

8

10

12

14

16

-0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10

Series: Residuals

Sample 1998M05 2009M09

Observations 137

Mean -3.85e-17

Median 0.006459

Maximum 0.129768

Minimum -0.247331

Std. Dev. 0.073638

Skewness -1.144578

Kurtosis 5.091726



0

4

8

12

16

20

-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08

Series: Residuals

Sample 1998M08 2009M09

Observations 134

Mean -5.27e-05

Median 0.001064

Maximum 0.075191

Minimum -0.064799

Std. Dev. 0.018890

Skewness -0.280145

Kurtosis 5.067134




Appendix 7: Engle-Granger Cointegration test

Money Market

Null Hypothesis: M_RESID has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.882910

10% level -2.578244


Cross Currency

Null Hypothesis: CC_RESID has a unit root

Exogenous: Constant


t-Statistic Prob.*



5% level -2.883073

10% level -2.578331



Appendix 8 Invalid ECM GARCH

Money Market Hedge


Method: ML - ARCH (Marquardt) - Normal distribution

Date: 05/29/15 Time: 00:28




Presample variance: backcast (parameter = 0.7)

GARCH = C(8) + C(9)*RESID(-1)^2 + C(10)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 4.35E-05 5.03E-05 0.865790 0.3866

DS1(-1) -0.178046 0.114602 -1.553597 0.1203

DS1(-2) -0.023720 0.098481 -0.240855 0.8097

DF 1.001371 0.000655 1528.940 0.0000

DF(-1) 0.178275 0.114673 1.554635 0.1200

DF(-2) 0.022908 0.098568 0.232413 0.8162

E(-1) -0.028378 0.016487 -1.721217 0.0852

Variance Equation

C 9.68E-09 3.69E-09 2.625780 0.0086

RESID(-1)^2 0.714328 0.224319 3.184424 0.0015

GARCH(-1) 0.245870 0.145005 1.695593 0.0900






Durbin-Watson stat 1.796215




Method: ML - ARCH (Marquardt) - Normal distribution

Date: 05/29/15 Time: 00:26



Convergence not achieved after 500 iterations

Presample variance: backcast (parameter = 0.7)

GARCH = C(8) + C(9)*RESID(-1)^2 + C(10)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.001799 0.003786 0.475119 0.6347

DS1(-1) -0.036657 0.122071 -0.300295 0.7640

DS1(-2) 0.105959 0.122128 0.867608 0.3856

DS2 0.501022 0.049486 10.12448 0.0000

DS2(-1) -0.073020 0.091664 -0.796599 0.4257

DS2(-2) 0.023119 0.080749 0.286305 0.7746

E2(-1) -254.2845 369.9638 -0.687323 0.4919

Variance Equation

C 2.46E-05 4.56E-05 0.539195 0.5898

RESID(-1)^2 0.139319 0.088195 1.579673 0.1142

GARCH(-1) 0.816629 0.208638 3.914103 0.0001






Durbin-Watson stat 1.737479

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Econometric Modeling in relation to Foreign Exchange Risk