Modeling and Forecasting Volatility in Foreign Exchange Markets

Sylwia Nowak

School of Economics, Australian National University

Sirimon Treepongkaruna*

School of Finance and Applied Statistics, Australian National University

Abstract

This study examines the properties of the GARCH model and its usefulness in modeling and

forecasting the volatility of exchange rate movements. The study adopts an approach to

forecasting out-of-sample that avoids any look-ahead bias. We find that the performance of

the forecast varies and depends upon the estimator of the latent volatility. An averaging

procedure produces superior forecasts compared to the standard practice of using daily point

estimates, however, the magnitude of the errors remains a concern irrespective of the method.

Moreover, across currencies and choice of metric, the out-of-sample analysis confirms how

difficult it is to forecast exchange rate movements.

JEL Classification: C220; F310; G150

Keywords: Foreign exchange rates; ARCH; financial time-series; volatility

* Corresponding author: College of Business and Economics, Australian National University, ACT 0200, Australia. Ph: +61-2-6125 3471, Fax: +61-2-6125 0087, sirimon.treepongkaruna@anu.edu.au.

2

Modeling and Forecasting Volatility in Foreign Exchange Markets

1. Introduction

Two assumptions generally made in the study of financial markets are that future returns are

independently and identically normally distributed and that their variance is constant through

the time. These assumptions conveniently lead to models of price movements that incorporate

only first-order moments. However, there is now a large volume of empirical evidence that

demonstrates that neither assumption generally holds in the context of risky assets. For

instance, empirical distributions of returns often exhibit leptokurtic properties. Further,

analysis of volatility typically shows that it varies through time and is prone to periods

of clustering. That is, large (small) changes in price to be followed by other large (small)

changes, in either direction (Bollerslev, 2001).

To account for these so-called ‘stylised facts’, researchers have incorporated time-varying

second-order moments into various models, popularly through the family of ARCH processes.

There has been considerable prior research that has examined the fit of ARCH processes in

relation to financial asset prices.1 Generally this research has attempted to find models of best

fit, with some research dedicated to an examination of the predictive ability of the various

models.

1 For comprehensive reviews of the literature, see Bollerslev et al. (1992) and Engle (1993 and 2002b).

3

For a time-series model to serve a useful purpose in practice, it should generally provide out-

of-sample forecasts that are of some predictive value. Volatility forecasts are particularly useful

in the investment and funds management industry for the purposes of portfolio construction,

security valuation and risk management; and the financial services industry for treasury

operations where estimates are required for managing portfolio positions through models such

as Value-at-Risk. Further, volatility is often a critical input for the valuation of derivative

instruments. However, there remain divergent views on the most appropriate model

of volatility. For example, Poon and Granger (2003), who provide a comprehensive review

of studies that have examined out-of-sample forecasting performance of various models

of volatility, conclude that “financial market volatility is clearly forecastable” but “as a rule

of thumb, historical volatility methods work equally well compared with more sophisticated

GARCH class and [stochastic volatility] models” (Poon and Granger, 2003). In contrast,

Ederington and Guan (2005) find that the GARCH model generally yields better forecasts

than the historic standard deviation and smoothing models. There are many other instances

where empirical findings have provided contradictory evidence.

This paper makes a further contribution to the literature by investigating the out-of-sample

predictive ability of the GARCH model in the context of foreign exchange markets. The data

cover a period of considerable change in these markets. The period studied is 1990 to 2004

which spans the introduction of the Euro, the move of several developing markets to open

economies, a significant depreciation on the US dollar and the Asian currency crisis of the late

1990s.

4

The purpose of this paper is to initially empirically assess the properties of foreign exchange

returns within the context of the applicability of GARCH models, and then attempt to forecast

the volatility of these returns in an out-of-sample exercise. Specifically, we examine the

forecasting power of the GARCH(1,1) model using time-series of daily spot exchange rates for

the Australian Dollar, British Sterling, French Franc, German Mark, Italian Lira, Japanese Yen,

Singaporean Dollar, Swiss Franc and Taiwan New Dollar. We demonstrate that the evaluation

of the out-of-sample predictions depends strongly upon the selection of the estimator for

latent volatility as well as the forecast error metric.

This remaining of this paper is organized as follows: Section 2 provides the background on the

models and their specification. Section 3 describes the data and the method. Section 4 presents

the empirical results while Section 5 concludes.

2. ARCH and GARCH models

Autoregressive Conditional Heteroscedasticity (ARCH) models, introduced by Engle (1982)

and generalised to GARCH by Bollerslev (1986), are models of conditional variance. In these

models, the variance is allowed to vary over time depending on the previous states of the

world (Bollerslev et al., 1994). The models represent an attempt to extract volatility from

historical observed returns. A stochastic process εt, parameterised by an unknown parameter(s)

5

θ, is said to follow a GARCH(p,q) process if its conditional mean equals zero conditioned on

the past information set It-1, and the conditional variance is specified as:2,3

( ) ∑∑=

−=

−−− σβ+εα+ω=ε≡σq

1i

2iti

p

1i

2iti1tt1t

2t I|Var t = 1, 2, … (1)

If all βi in (1) are zero, then the variance collapses to a function of the weighted average of past

squared innovations, as in the original ARCH model developed by Engle (1983).

The conditional distribution of εt is usually assumed to be normal (Bollerslev et al., 1994),

which is consistent with a heavy-tailed unconditional distribution.4

The simple GARCH(1,1) process has gained popularity and is perceived as a model that is

capable of capturing volatility clustering across various financial asset returns. As noted by

Engle, the GARCH(1,1) model is considered to be “a generally excellent model for a wide

2 Throughout this paper the dependence of tε and tσ on the parameter vector θ are suppressed for the

convenience of notation.

3 Only few processes have a constant mean of zero, so usually tε will be defined as the residual term from some

model, viz: γXYε 'ttt −= (Bollerslev et al., 1994).

4 As Bollerslev et al. (1994) note, “the degree of leptokurtosis induced by the time-varying conditional variance

often does not capture all of the leptokurtosis present in high frequency speculative process.” Thus, extensions

to ARCH andGARCH models have been proposed that allow for alternate conditional distributions including

the Student’s t-distribution (Bollerslev, 1987), power exponential distribution (Baillie and Bollerslev, 1989),

generalised error distribution (Nelson 1991) and normal inverse Gaussian (Barndorff and Nielsen, 1997).

6

range of financial data” (Engle, 1993). Yet numerous sophisticated modifications to the basic

model are available and these have tended to follow stylized facts observed in time-series of

financial returns.5 But while the ability of the GARCH models to fit the in-sample data well is

commonly praised, the out-of-sample forecasting power of these models remains questionable.

There have been a few attempts to forecast out-of-sample volatility but their conclusions vary

dramatically. For instance, ARCH models have been found to be an excellent forecasting tool

(Akgiray, 1989); but have also been reported to perform indifferently (Brailsford and Faff,

1996); and have been found to perform worse than a simple linear regression (Dimson and

Marsh, 1990).

The problem with forecasting volatility starts with the fact that the true underlying volatility is

unobservable. A common approach to estimating a proxy for ex-post volatility is to use the

squared return innovation over the relevant horizon. However, while the squared innovation

provides an unbiased estimate for the latent volatility factor, it can yield very noisy

measurements (Andersen and Bollerslev, 1998). Hence to eliminate noise, the more frequent

squared innovations 2tε are usually accumulated over some longer period to estimate the less

frequent latent volatility. For example, cumulative squared intra-day innovations are used to

estimate daily variances (Andersen and Bollerslev, 1998), or cumulative squared daily

innovations are used to estimate monthly variances (Figlewski, 1997).

5 An informative discussion of stylized facts and how they motivated researchers to examine new models can be

found in Bollerslev et al. (1994).

7

The other difficulty is that models generally “focus on variance one step ahead. They are not

designed to produce variance forecasts for a long horizon” (Figlewski, 1997). Thus, if there is

no new information, the series will converge to the long-run variance. Under the GARCH(1,1)

model, the forecast series can be obtained by:

21t

21t

2t −− βε+ασ+ω=σ

( ) ( ) ( ) 2t

2tt

21t

21tt EE σβ+α+ω=εβ+ασ+ω=σ −+

M

( ) ( ) ( ) 2t

k1k

0s

s2kttE σβ+α+β+αω=σ ∑

−

=+ (2)

From (2), it is clear that as the length of the forecast period increases, the model is likely to

become less accurate as there is no update of information. Thus, in practice, the model

requires re-estimation of the parameters with contemporary data to update information and

this can be cumbersome to apply as the update requires a sequence of models to be fitted over

a rolling sample of returns.

3. Data and Method

3.1. Data

The data consist of daily spot exchange rates for the currencies of Australia (AUD), France

(FRF), (West) Germany (DEM), Italy (ITL), Japan (JPY), Switzerland (CHF), Singapore

(SHD), Taiwan (TWD) and United Kingdom (GBP). These rates are all measured against the

8

US Dollar.6 The daily mid-rates were collected from Datastream for the period from 1 January

1990 to 30 July 2004. The sample comprises 3805 observations for each series (excluding

weekends). Plots of the data are presented in Figure 1. Consistent with prior literature, each

currency series follows a stochastic trend and exhibits no clear pattern.

[Figure 1 about here]

The sample period is partitioned into a twelve-year in-sample estimation period from 1 January

1990 to 31 December 2001 (3131 observations) and a subsequent out-of-sample forecasting

period from 1 January 2002 to 30 July 2004 (674 observations). The partitioning of the series,

though purely arbitrary, allows for a long estimation period as GARCH models rely upon a

relatively large number of data points for robust estimation. Of note, the in-sample period

includes the Asian financial crisis in 1997 and the introduction of the Euro in 1999, whereas

the out-of-sample period covers a strong depreciation of the US Dollar and the aftermath of

the events of September 11. Thus, the in-sample and out-of-sample periods cover different

economic conditions and this adds to the complexity of the forecast challenge.

Following Bollerslev et al. (1994), nominal percentage returns are computed, viz.:

( ) ( )[ ]1ttt YlnYln100X −−⋅≡ (3)

6 The rates for the ‘old’ European Union currencies (i.e. French Franc, German Mark and Italian Lira) were

obtained from Datastream and converted at the rates set by the European Central Bank on 31 December 1998.

9

where Yt is the mid-rate of the relevant exchange rate series relative to the US Dollar on day t.

It is a well-known stylized fact that exchange rate series have a unit root, whereby first

differences in logarithms of the spot rates are stationary. Confirmatory results are obtained for

each of the nine currencies under examination in this study. The results of the augmented

Dickey-Fuller (1979) tests for stationarity, with the null hypothesis of unit root, are presented

in Table 1. The table highlights that the nominal percentage returns (Xt) are stationary,7

whereas the raw series (Yt) is not stationary whether measured in levels or logarithmic scale.

[Table 1 about here]

Table 2 provides summary statistics of the (nominal) exchange rate returns. Separate sets

of descriptive statistics are presented for the in-sample, out-of-sample and full sample periods

(Panels A, B and C, respectively). The average return over the in-sample period is positive for

all currencies except Japan and Singapore (even though all median returns are zero).

In contrast, the average return over the out-of-sample period is negative for all currencies. This

feature highlights the differences between the in-sample of out-of-sample periods. Moreover,

even though the in-sample data set is almost five times larger than the out-of-sample data set,

the average return over the full sample period tends to be swamped by the smaller out-of-

sample period as indicated by the negative average return over the full period for most 7 A variable is said to be weakly stationary if the mean and autocovariances of the series are independent of time

(Brockwell and Davis, 2002).

10

currencies. Similarly, the various exchange rate returns tend to exhibit a tendency for negative

skewness over the in-sample period and positive skewness over the out-of-sample period.

[Table 2 about here]

Figure 2 plots the daily exchange rate returns. From this figure, there is visual evidence of

heteroscedastic attributes and the clustering of volatility can be seen. Further, leptokurtosis can

be seen in Figure 3. This figure contains the normal quantile-quantile plots for the daily

exchange rate returns. The plots posses the characteristic S-shape indicating that there is no

significant skewness, but the tails are heavier than a normal distribution (Andersen et al., 2000).

Overall, the data under analysis exhibit the common features consistent with the stylized facts

of exchange rate series previously documented (DeVries, 1994).

[Figure 2 about here]

[Figure 3 about here]

As discussed by Baillie and Bollerslev (1989), the fact that the unit root hypothesis cannot be

rejected for any of the currencies implies that the random-walk model as an explanation of rate

movements is inappropriate. Researchers have consequently looked for alternate time-series

models such as the ARCH family of models (Baillie and Bollerslev, 1989).8

8 Note that the method adopted herein implicitly treats each exchange rate as an independent series. In reality,

exchange rates are correlated and thus there is some argument for adopting a multivariate approach. However,

11

3.2. Estimation Method

Following Bollerslev et al. (1994), the conditional variance of each series is initially modeled

using an MA(1) – GARCH(1,1) process, viz:9

t1ttX ε+θε+μ= − (4)

t2

1t2

1t2t ν+βσ+αε+ω=σ −− (5)

Before the models are estimated, Engle’s (1983) test for ARCH effects is performed. Under

the null hypothesis, the variance from (5) is simply a constant and therefore the model (4) can

be estimated using ordinary least squares and the Lagrange multiplier test for p-th order

heteroscedasticity can be applied, as proposed by Breusch and Pagan (1980). Auxiliary

regressions are run of the squared OLS residuals 2tε , obtained from (4), on a constant and

squared p lags: 2pt

21t ε,...,ε −− , where p = 1, 4, 8. The number of observations (N-p) multiplied by

the R-squared of the auxiliary regression follows a chi-squared distribution with p degrees of

freedom under the null hypothesis of homoscedasticity.

as the focus in the paper is on forecasting each exchange rate and as they are separately examined, we leave

a multivariate approach for future work.

9 A set of day-of-week dummy variables was augmented to (4), to avoid any bias arising from the so-called

weekend effect (Baillie and Bollerslev, 1989). However, all coefficients on the day-of-week dummy variables

were insignificant. These results are not presented here but can be obtained from the authors.

12

The results of the tests for ARCH effects are provided in Table 3. The null hypothesis of white

noise errors is rejected in all cases at the 1% significance level for the in-sample period (Panel

A) and the full sample period (Panel C), consistent with a heteroscedastic structure in the

residuals. However, in the case of the out-of-sample period (Panel B), there is mixed evidence

of the rejection of the null hypothesis of homoscedasticity. In this period, the null hypothesis

is rejected at 8 lags for all series except Japan and Singapore, but the null hypothesis cannot

generally be rejected at lower lags. These mixed results may be due to the relatively short

length of the out-of-sample period, but are also consistent with earlier evidence that the in-

sample and out-of-sample periods exhibit different attributes.

[Table 3 about here]

3.3. Forecasting technique

Given the nature of the GARCH model, better forecasts should be attained when more

frequent squared innovations 2tε are cumulated to estimate less frequent (latent) volatility.

Therefore, we need to specify a period over which the individual forecasts are cumulated to

form the base for the evaluation. In the absence of any more logical frequency, we select

monthly intervals as the relevant evaluation frequency. Thus, we require monthly volatility

sequences for the out-of-sample period (1 January 2002 – 30 July 2004). We employ two

alternative procedures.

13

The first forecast series (F1) are obtained by employing the following procedure.10 First, using

parameter estimates obtained from the in-sample data, daily k-step ahead volatility forecasts

2stσ̂ + are generated using (2) for all trading days in the first month of the out-of-sample period.

Then, the monthly volatility forecast is formed by summing the k-step ahead daily forecasts,

i.e.:

∑=

+σ=σTN

1s

2st

2T ˆˆ (6)

Hence, initially we generate a set of daily forecasts for each trading day in January 2002 that

provides the monthly forecast for January 2002. The application of a rolling window then

moves the start and end dates of the in-sample estimation period forward one month such that

observations from January 1990 are deleted and observations from January 2002 are added to

the (in-sample) estimation period, and the estimation window remains at a constant length of

144 months. The procedure described above is then repeated to produce a volatility forecast

for February 2002. The window is then rolled forward one month at a time until a volatility

estimate is produced for July 2004 which is the last month in the out-of-sample period. Thus,

there are 31 forecasts that form the evaluation set over the period January 2002 to July 2004.

This consequent series of forecasts is based solely on ex-post data and there is no look-forward

bias in the forecasts. To this extent, the forecast method employed herein is capable of being

applied in practice.

10 A similar procedure has been used by Brailsford and Faff (1996), Wang and Wong (1997) and Figlewski (1997).

14

The second forecast series (F2) is similar except that we employ a different starting value for

the initial volatility point estimate. The k-step ahead daily forecasts under the GARCH model

as in (2), rely on both the parameter estimates and the initial point estimate of conditional

volatility which is itself reliant upon past innovations. Thus, if there is a consistent bias in the

innovations at the end of each period (month) then this will feed into the forecasts of the next

period (month). Moreover, in a small number of forecasts, spurious results may be obtained by

a few extreme end-of-period (month) values. Thus, we employ the average of the previous

month’s daily innovations in the initial estimate of the conditional variance, viz:

∑=

ε⋅=σTN

1t

2t

T

2t

N1 (7)

The conditional variance from (7) is used with the same parameter estimates as the first

method to produce daily k-step ahead volatility forecasts 2stσ̂ + using (2) for all trading days in

the first month of the out-of-sample period. Then, the monthly volatility forecast is formed by

summing the k-step ahead daily forecasts as per (6). The rolling window approach is again

applied to produce the series of monthly volatility forecasts.

Hence, the only difference between the two forecast methods is the starting estimate of the

conditional variance. As the second series (F2) relies upon an average value, a priori we expect

this forecast series to be smoother than the first series (F1) although we have no priors as to

how this will impact on forecast accuracy.

15

The forecasts need to be evaluated against actual volatility, however this variable is

unobservable. Previous studies have used a variety of proxies with the two main approaches

involving either realized volatility estimated from squared returns, or implied volatility reverse

engineered from option prices. The latter approach involves a joint assumption about market

efficiency and the validity of the pricing model which we wish to avoid. Moreover, implied

volatility itself has been used as the forecast estimator rather than the benchmark (eg. Day and

Lewis, 1992; Xu and Taylor, 1995). Hence we utilize observed returns to generate realized

volatility and construct the monthly volatility point by summation of the daily squared returns

over NT trading days in any month T, viz:

∑=

=σTN

1t

2t

2T X (8)

4. Empirical Results

4.1. Coefficient Estimates

Table 4 reports the estimated coefficients of the parameters in (4) and (5) together with their

robust standard errors computed using quasi-maximum likelihood methods (Bollerslev and

Wooldridge, 1992).11 From Table 4, the coefficient estimates of the conditional mean are all

insignificantly different from zero for all series. In contrast, the GARCH effects are significant 11 Note that the GARCH parameter estimates remain consistent even if the conditional normality assumption is

violated (providing the mean and variance functions are correctly specified), however, their standard errors will

be incorrect. Hence, suspecting non-normality, quasi-maximum likelihood estimators are used (Bollerslev and

Wooldridge, 1992).

16

as indicated by the highly significant coefficient estimates on α and β, providing further

support for the adoption of the model (at least for the in-sample period). Note that the sum of

the estimated GARCH parameters is close to the unity in all cases, showing a strong degree of

persistence in the conditional variance. 12 As noted by Anderson and Bollerslev (1989), this

suggests that “financial market volatility is highly predictable” so one might expect reasonably

accurate forecasts. However, the coefficients vary noticeably between the in-sample and out-

of-sample data, and it is this variation that makes point estimates difficult to predict.

[Table 4 about here]

Given the results in Table 4, and conscious of over-parameterisation, the MA(1) element is

dropped from the conditional mean equation in (4) and a more parsimonious model is re-

estimated.13 Moreover, as the conditional mean is insignificantly different from zero, the mean

of each series is assumed to be zero and therefore the estimation model collapses to the

conditional variance, viz:14,15

12 In such a situation, the Integrated GARCH model of Engle and Bollerslev (1986) could be employed to model

long-run volatility. However, this study is concerned with short-term volatility and thus avoids the use of

IGARCH (Andersen and Bollerslev, 1989).

13 These results are not presented here but can be obtained from the authors.

14 A similar approach is employed elsewhere (Brailsford and Faff, 1996; Andersen and Bollerslev, 1998);

Figlewski, 1997). These previous studies all report that the coefficients in the mean equation are insignificant,

and Figlewski (1997) finds that the assumption of a zero mean improves the out-of-sample forecast accuracy.

17

t2

1t2

1t2t X ν+βσ+α+ω=σ −− (9)

4.2. Forecast Results

Figure 4 plots the forecast and realized (monthly) volatility series for each of the nine

currencies over the out-of-sample period. A quick visual inspection reveals that the forecasts

from the second method (F2) that use an average as the initial estimate of conditional variance

each period provide a more accurate mapping of the realized volatility. Moreover, there

appears to be a general common trend between the forecast values and the realized volatility,

except in the case of Taiwan.

[Figure 4 about here]

However, Figure 4 provides only visual evidence and it is difficult to draw conclusions. Hence,

we turn to a more formal evaluation using error metrics to assess point estimates. A variety of

error metrics are available and have been previously used (see Brailsford and Faff, 1996; Byers

and Nowman, 1998). We select some common metrics that include the mean error (ME),

mean absolute error (MAE), root mean squared error (RMSE), mean absolute percentage error

(MAPE) and Theil’s inequity measure (U). These error metrics are defined as follows:

15 Note that a consequence of the forecast model relying only on the conditional variance as in (9) is that the

innovations tε in (7) become substituted by the returns, Xt.

18

( )∑=

−=KN

1T

2T

2T

K

σσ̂N1ME (10)

∑=

−=KN

1T

2T

2T

K

σσ̂N1MAE (11)

( )∑=

−=KN

1T

22T

2T

K

σσ̂N1RMSE (12)

( )∑=

−=KN

1T

2T

2T

2T

K

σσσ̂N1MAPE (13)

( )

( ) ( )∑∑

∑

==

=

+

−=

KK

K

N

1T

22T

K

N

1T

22T

K

N

1T

22T

2T

K

σN1σ̂

N1

σσ̂N1

U (14)

where NK is the total number of forecasts, i.e. 31.

The results of the application of the error metrics are reported in Table 5. While it is difficult

to draw an overall conclusion given the large number of results arising from multiple series and

multiple error metrics, the second method (F2) generally provides more accurate forecasts.

However, it should be noted that the evaluation of the forecast does vary a little dependent

upon the choice of the error metric. Nonetheless, there is a reasonably high degree of

consistency across the metrics that provide lower values for the second method across all

currencies. The difference in the two methods highlights a problem of forecasting with

GARCH especially over short periods, and the sensitivity of estimates to the starting values of

conditional variance.

19

[Table 5 about here]

We do not place much weight on the mean error (ME) in Table 5 as it allows the errors of

opposite signs to cancel each other. However, this metric does indicate whether volatility is

under- or over-predicted. From Table 5, the first method (F1) under-predicts the volatility of

Australian Dollar, Swiss Frank, German Mark, British Pound and Singapore Dollar and over-

predicts the remaining currencies, with the Yen giving rise to a notably large mean error. In

contrast, the second method (F2) only under-predicts two currencies being the Australian

Dollar and German Mark. In this case, the Taiwanese New Dollar is associated with a notably

large mean error.

The MAE presents a sobering set of statistics with both models exhibiting relatively high

measures. The second method (F2) is again preferred generally yielding error values around

half of those from the first method (F1). The worst cases appear to be the Australian Dollar

and Swiss Franc and the best cases are the Singapore Dollar and British Pound. The RMSE

metric provides a similar story.

Turning to percentage errors in the MAPE metric, which is arguably the most relevant for

practitioners, the second method (F2) is again generally preferred. However, when looking at

the individual currencies, the most accurate forecasts are obtained for the French Franc and by

far the worst are obtained for the Taiwan New Dollar. Theil’s statistic, which is bounded

between 0 and 1, is relatively low for all currencies using the second method (F2), with the

20

forecasts for the British Pound being the most accurate and for the Taiwan New Dollar being

the least accurate by a considerable margin. Also of note is that despite the fact that we have

used a GARCH model, the currencies that are the most difficult to forecast are those that

exhibited the largest levels of unconditional kurtosis.

The error metrics generally support the use of the second method (F2). Given that the two

forecast series differ only by the selection of the initial estimate of conditional variance each

period, it highlights the sensitivity of forecasts from the GARCH model as in (2) to these

initial estimates. The second method relies upon an averaging procedure over the prior month

which has the effect of smoothing the initial estimate across months and consequently

producing a smoother series of monthly forecasts. However, this does not a priori lead to

superior forecasts per se, however notwithstanding the results herein do support this method

as yielding superior forecasts.

5. Summary

This paper has examined the properties of the ARCH and GARCH model and their usefulness

in modeling and predicting the volatility of exchange rate returns. Initial examination of

foreign exchange rates from nine currencies with varying underlying economies revealed

ARCH effects and subsequent modelling using the GARCH model was undertaken. This

model was then used to generate out-of-sample volatility forecasts which were evaluated

against realized volatility. The approach was carefully constructed so to avoid any look-ahead

bias. Two series of forecasts were produced that varied through the initial estimate of

21

conditional variance for each period – an end-of-month point estimate and a month average

estimate. Through analysis of a series of error metrics, the method that utilizes the average of

the previous month was preferred, but this nonetheless generated errors of magnitude that

would be most likely be of concern to practitioners. While the results vary across currencies

and choice of metric, the out-of-sample analysis confirms how difficult it is to forecast

exchange rate movements. This is particularly so over the last few years in the Asian

currencies. A consequent message is that exposure to exchange rates must come with the

realisation that their unpredictable nature means that unpredictable gains and losses will be

incurred. As such, fund managers with international holdings must consider whether to hedge

their underlying returns or remain exposed to unpredictable exchange rate movements.

22

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27

Table 1 The Augmented Dickey-Fuller Test for Stationarity in Daily Exchange Rates

This table presents the results of the augmented Dickey-Fuller (1979) tests for stationarity of Australian Dollar (AUD), Swiss Franc (CHF), German Mark (DEM), French Franc (FRF), British Pound (GBP), Italian Lira (ITL), Japanese Yen (JPY), Singaporean Dollar (SHD) and Taiwan New Dollar (TWD) vs the US Dollar daily exchange rates. These are the daily mid-rates for the period from 1 January 1990 to 30 July 2004 for a total of 3805 observations excluding weekends. The tests, with the null hypothesis of unit root, are performed on the level and the logarithmic scales of the exchange rates directly as well as on the nominal percentage returns, calculated as (Bollerslev, Engle and Nelson, 1994):

( ) ( )[ ]1ttt YlnYln100X −−⋅≡

Auxiliary regressions are of the form: − t1t21t10t εYΔφYφφYΔ +++= −− for the level rates, − ( ) ( ) ( ) t1t21t10t εYlogΔφYlogφφYlogΔ +++= −− for the logarithms of the rates, − t1t0t εXφXΔ += − for the returns.

AUD CHF DEM FRF GBP ITL JPY SHD TWD

Exchange Rate Yt -1.5646 -2.0302 -1.4851 -1.5265 -2.0762 -1.4650 -2.3663 -1.8932 -0.7746 log(Yt) -1.5916 -2.0505 -1.5483 -1.5837 -2.0765 -1.4912 -2.2710 -1.7636 -0.8132

Return Xt -62.7691* -63.5218* -63.1896* -61.9879* -62.4215* -61.3127* -62.6009* -64.3975* -83.4224*

* denotes that the null of unit root is rejected at 1% significance level.

28

Table 2 Descriptive Statistics of Daily Exchange Rate Returns

This table presents the summary statistics of the daily nominal percentage exchange rate returns for the Australian Dollar (AUD), Swiss Franc (CHF), German Mark (DEM), French Franc (FRF), British Pound (GBP), Italian Lira (ITL), Japanese Yen (JPY), Singaporean Dollar (SHD) and Taiwan New Dollar (TWD) vs the US Dollar. Panel A presents the statistics for the in-sample period from 2 January 1990 to 31 December 2001 (3130 observations), Panel B the statistics for the out-of-sample period from 1 January 2002 to 30 July 2004 (674 observations). Panel C presents the statistics for the full sample period from 2 January 1990 to 30 July 2004 (3804 observations).

AUD CHF DEM FRF GBP ITL JPY SHD TWD

Panel A: 02 January 1990 – 31 December 2001

Mean 0.0138 0.0024 0.0084 0.0077 0.0033 0.0172 -0.0030 -0.0009 0.0094 Median 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Std. Dev. 0.6131 0.7443 0.6828 0.6547 0.6138 0.6596 0.7379 0.3536 0.5137 Skewness -0.1291 -0.1349 -0.0521 -0.0636 0.1260 0.6687 -0.8686 -1.1300 0.0084Kurtosis 7.6403 4.9740 4.9420 5.7638 7.4068 10.0827 11.1978 24.618 14.651

Panel B: 01 January 2002 – 30 July 2004

Mean -0.0471 -0.0384 -0.0445 -0.0445 -0.0331 -0.0445 -0.0241 -0.0105 -0.0043

Median -0.0771 -0.0310 -0.0392 -0.0320 -0.0467 -0.0303 -0.0371 0.0000 0.0000Std. Dev. 0.7005 0.7122 0.6482 0.6225 0.5340 0.6232 0.5899 0.2863 0.2046 Skewness 0.5081 0.0535 0.1488 0.3326 0.2269 0.3339 0.1040 0.0908 0.8386 Kurtosis 4.4619 3.6068 3.4201 3.5277 3.9454 3.5311 4.0460 5.9674 8.3204

Panel C: 02 January 1990 – 30 July 2004

Mean 0.0031 -0.0049 -0.0010 -0.0015 -0.0032 0.0063 -0.0067 -0.0026 0.0070

Median 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 Std. Dev. 0.6298 0.7388 0.6770 0.6493 0.6005 0.6536 0.7139 0.3426 0.4739 Skewness 0.0133 -0.1029 -0.0175 0.0016 0.1456 0.6210 -0.7738 -1.0081 0.0332 Kurtosis 6.8158 4.7629 4.7113 5.4156 7.0882 9.1463 10.8321 23.470 16.697

29

Table 3 Tests for Heteroscedasticity in Daily Exchange Rates

This table presents the results of tests for ARCH effects, as proposed by Engle (1983), in daily nominal percentage exchange rate returns for the Australian Dollar (AUD), Swiss Franc (CHF), German Mark (DEM), French Franc (FRF), British Pound (GBP), Italian Lira (ITL), Japanese Yen (JPY), Singaporean Dollar (SHD) and Taiwan New Dollar (TWD) vs the US Dollar. Panel A presents the statistics for the in-sample period from 2 January 1990 to 31 December 2001 (3130 observations), Panel B the statistics for the out-of-sample period from 1 January 2002 to 30 July 2004 (674 observations). Panel C presents the statistics for the full sample period from 2 January 1990 to 30 July 2004 (3804 observations). The data are modeled using MA(1) – GARCH(1,1) process (Bollerslev et al., 1994), i.e.: t1tt εθεμX ++= − and t

21t

21t

2t νβσαεωσ +++= −− . Under the null hypothesis,

the variance 2tσ is a constant so the mean equation can be estimated using ordinary least squares and the Lagrange

multiplier test for p-th order heteroscedasticity can be applied, as proposed by Breusch and Pagan (1980). The auxiliary regressions are run on the squared OLS residuals 2

tε on a constant and squared p lags: 2pt

21t ε,...,ε −− , where

8,4,1p = . The number of observations (N - p) multiplied by the R-squared of the auxiliary regression follows a chi-squared distribution with p degrees of freedom under the null.

Test AUD CHF DEM FRF GBP ITL JPY SHD TWD

Panel A: 02 January 1990 – 31 December 2001

p=1 13.4441* 22.3214* 20.6166* 79.2260* 59.9106* 15.2240* 37.1575* 139.5807* 304.4138*

p=4 45.9187* 36.7563* 50.2281* 138.1219* 125.1219* 282.6219* 51.4313* 375.0906* 335.1719*

p=8 57.4860* 66.0092* 71.8423* 280.3438* 156.4120* 295.9301* 62.9381* 467.9690* 351.4933*

Panel B: 01 January 2002 – 30 July 2004

p=1 10.8730* 0.0007 0.8360 0.0679 1.7459 0.0564 4.0730** 0.3178 20.3038*

p=4 16.9772* 5.2670 8.4729 6.1796 15.5757* 6.1153 6.4953 1.0684 23.7187*

p=8 25.9922* 13.6637 23.6434* 19.8077** 26.2908* 19.8529** 12.0325 14.0268 92.4842*

Panel C: 02 January 1990 – 30 July 2004

p=1 22.4850* 24.2682* 18.3561* 78.7622* 69.1660* 16.2041* 46.0422* 167.2234* 380.2000*

p=4 57.3041* 40.4290* 54.7398* 147.3518* 151.8460* 316.1908* 64.2183* 442.3176* 422.9085*

p=8 76.0025* 73.3422* 81.1447* 304.1085* 189.6020* 329.4591* 77.8582* 554.5406* 445.4419*

* denotes that the null hypothesis of homoscedasticity is rejected at 1% significance level. ** denotes that the null hypothesis of homoscedasticity is rejected at 5% significance level.

30

Table 4 GARCH Model Parameters Estimates from Daily Exchange Rate Returns

This table presents the parameter estimates from fitting an MA(1) – GARCH(1,1) process (Bollerslev et al., 1994), i.e.: t1tt εθεμX ++= − and t

21t

21t

2t νβσαεωσ +++= −− . The model is applied to daily nominal percentage exchange

rate returns for the Australian Dollar (AUD), Swiss Franc (CHF), German Mark (DEM), French Franc (FRF), British Pound (GBP), Italian Lira (ITL), Japanese Yen (JPY), Singaporean Dollar (SHD) and Taiwan New Dollar (TWD) vs the US Dollar over the in-sample period from 2 January 1990 to 31 December 2001 (3130 observations). Robust standard errors are reported under the parameter estimates, and are computed using quasi-maximum likelihood methods (Bollerslev and Wooldridge, 1992). AUD CHF DEM FRF GBP ITL JPY SHD TWD

μ 0.0083 0.0059 0.0092 0.0045 -0.0064 0.0031 0.0045 -0.0078 0.0003 0.0100 0.0125 0.0113 0.0111 0.0098 0.0106 0.0119 0.0037** 0.0058 θ -0.0134 -0.0068 -0.0128 0.0186 -0.0159 0.0210 -0.0249 0.0153 -0.1890 0.0202 0.0190 0.0189 0.0191 0.0197 0.0205 0.0214 0.0217 0.0299* ω 0.0012 0.0124 0.0058 0.0054 0.0042 0.0039 0.0079 0.0009 0.0063 0.0010 0.0051* 0.0021* 0.0020* 0.0016* 0.0016** 0.0038** 0.0004* 0.0029**

α 0.0215 0.0315 0.0343 0.0389 0.0436 0.0555 0.0374 0.0858 0.2200 0.0052* 0.0078* 0.0069* 0.0079* 0.0118* 0.0144* 0.0115* 0.0180* 0.0460* β 0.9754 0.9455 0.9530 0.9482 0.9451 0.9371 0.9485 0.9089 0.7995 0.0072* 0.0139* 0.0084* 0.0097* 0.0135* 0.0150* 0.0152* 0.0213* 0.0355*

* denotes that a coefficient is significant at 1% significance level. ** denotes that a coefficient is significant at 5% significance level.

31

Table 5 Error Metrics from Forecasting Monthly Volatility using Out-of-Sample Exchange

Rate Returns

This table reports the error metrics from evaluating out-of-sample forecast series generated from the GARCH(1,1) model based on nominal percentage exchange rate returns statistics benchmarked against realized volatility for the French Franc (FRF), German Mark (DEM), Italian Lira (ITL), Japanese Yen (JPY), Swiss Franc (CHF) and British Pound (GBP) vs the US Dollar for the out-of-sample period from 1 January 2002 to 30 July 2004. The forecasts are of monthly volatility (31 estimates). Two forecast series are examined with the variation driven by the estimate of the initial conditional variance each period. The reported error metrics are the mean

error: ( )∑=

−=31

1T

2T

2T σσ̂

311ME , the mean absolute error : ∑

=

−=31

1T

2T

2T σσ̂

311MAE , the root mean squared

error: ( )∑=

−=31

1T

22T

2T σσ̂

311RMSE , the mean absolute percentage error: ( )∑

=

−=31

1T

2T

2T

2T σσσ̂

311MAPE , and

Theil’s inequity measure: ( )

( ) ( )∑∑

∑

==

=

+

−=

31

1T

22T

31

1T

22T

31

1T

22T

2T

σ311σ̂

311

σσ̂311

U

Variable AUD CHF DEM FRF GBP ITL JPY SHD TWD

Forecast Series F1

ME -0.000388 -0.000113 -0.000086 0.000024 -0.000136 0.000036 0.000293 -0.000037 0.000153MAE 0.000813 0.000921 0.000675 0.000700 0.000381 0.000729 0.000799 0.000160 0.000170RMSE 0.001091 0.001126 0.000854 0.000940 0.000523 0.000991 0.001264 0.000212 0.000192MAPE 0.007350 0.008595 0.007861 0.009138 0.005649 0.009472 0.012039 0.009128 0.054687

U 0.004434 0.004323 0.003975 0.004463 0.003803 0.004595 0.005247 0.004889 0.005005

Forecast Series F2

ME -0.000026 0.000013 -0.000002 0.000003 0.000004 0.000018 0.000057 0.000013 0.000207MAE 0.000437 0.000433 0.000373 0.000319 0.000206 0.000332 0.000297 0.000085 0.000209RMSE 0.000612 0.000593 0.000485 0.000399 0.000296 0.000413 0.000361 0.000126 0.000234MAPE 0.005086 0.004882 0.004487 0.003973 0.004182 0.004113 0.004821 0.004809 0.063197

U 0.002474 0.002463 0.002437 0.002188 0.002091 0.002234 0.002147 0.003000 0.005292

32

Figure 1 Daily Exchange Rates vs US Dollar 1990-2004

This figure plots the series of daily exchange mid-rates for the Australian Dollar (AUD), Swiss Franc (CHF), German Mark (DEM), French Franc (FRF), British Pound (GBP), Italian Lira (ITL), Japanese Yen (JPY), Singaporean Dollar (SHD) and Taiwan New Dollar (TWD) vs the US Dollar over the period 2 January 1990 to 30 July 2004 for a total of 3805 observations excluding weekends.

1.0

1.2

1.4

1.6

1.8

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2.2

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

AUD

1.1

1.2

1.3

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90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

CHF

1.2

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2.4

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

DEM

4.5

5.0

5.5

6.0

6.5

7.0

7.5

8.0

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

FRF

.45

.50

.55

.60

.65

.70

.75

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

GBP

1000

1200

1400

1600

1800

2000

2200

2400

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

ITL

60

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100

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180

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

JPY

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2.0

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

SHD

24

26

28

30

32

34

36

90 91 92 93 94 95 96 97 98 99 00 01 02 03 04

TWD

33

Figure 2 Volatility of Daily Exchange Rate Returns 1990-2004

This figure plots the daily nominal percentage exchange rate returns for the French Franc (FRF), German Mark (DEM), Italian Lira (ITL), Japanese Yen (JPY), Swiss Franc (CHF) and British Pound (GBP) vs the US Dollar for the period from 2 January 1990 to 30 July 2004 (3804 observations).

-6

-4

-2

0

2

4

1990 1992 1994 1996 1998 2000 2002 2004

AUD

-4

-3

-2

-1

0

1

2

3

4

1990 1992 1994 1996 1998 2000 2002 2004

CHF

-4

-3

-2

-1

0

1

2

3

4

1990 1992 1994 1996 1998 2000 2002 2004

DEM

-5

-4

-3

-2

-1

0

1

2

3

4

1990 1992 1994 1996 1998 2000 2002 2004

FRF

-6

-4

-2

0

2

4

6

1990 1992 1994 1996 1998 2000 2002 2004

GBP

-4

-2

0

2

4

6

8

1990 1992 1994 1996 1998 2000 2002 2004

ITL

-8

-6

-4

-2

0

2

4

6

1990 1992 1994 1996 1998 2000 2002 2004

JPY

-5

-4

-3

-2

-1

0

1

2

3

1990 1992 1994 1996 1998 2000 2002 2004

SHD

-5

-4

-3

-2

-1

0

1

2

3

4

1990 1992 1994 1996 1998 2000 2002 2004

TWD

34

Figure 3 Quantile-Quantile Plots of Daily Exchange Rate Returns

This figure presents the normal quantile-quantile plots for the daily nominal percentage exchange rate returns for the French Franc (FRF), German Mark (DEM), Italian Lira (ITL), Japanese Yen (JPY), Swiss Franc (CHF) and British Pound (GBP) vs the US Dollar for the period from 2 January 1990 to 30 July 2004 (3804 observations).

Normal Quantiles

Qua

ntile

s of

AU

D

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

CH

F

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

DE

M

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

FR

F

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

GB

P

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

ITL

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

JP

Y

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

SH

D

-2 0 2

-8-6

-4-2

02

46

Normal Quantiles

Qua

ntile

s of

TW

D

-2 0 2

-8-6

-4-2

02

46

Figure 4 Comparison of Realized and Forecast Volatility of Monthly Exchange Rate Returns

This figure presents a comparison of the realized volatility with out-of-sample forecast series generated from the GARCH(1,1) model based on nominal percentage exchange rate returns statistics benchmarked against realized volatility for the French Franc (FRF), German Mark (DEM), Italian Lira (ITL), Japanese Yen (JPY), Swiss Franc (CHF) and British Pound (GBP) vs the US Dollar for the out-of-sample period from 1 January 2002 to 30 July 2004. The forecasts are of monthly volatility (31 estimates). Two forecast series are examined with the variation driven by the estimate of the initial conditional variance each period. The dotted line graphs forecast series F1; the dashed line graphs the forecast series F2; the solid line graphs the corresponding realized volatility.

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

510

1520

25

AUD

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

510

1520

CHF

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

510

15

DEM

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

48

1216

FRFV

olat

ility

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

26

1014

GBP

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

48

1216

ITL

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

46

812

16

JPY

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

12

34

5

SHD

Vol

atili

ty

2001Dec

2002Jun

2002Dec

2003Jun

2003Dec

2004Jun

01

23

4

TWD