Applied Einancial Economics, 2000, 10, 371-377
Forward foreign exchange rates and
expected future spot rates
CHRISTIAN C. P. WOLFF
Limburg Institute of Einancial Economics (LIFE). Maastricht University, P.O. Box
616. 6200 MD. Maastricht, The Netherlands and Centre for Economic Policy
Re.search (CEPRJ, London, UK
E-mail: [email protected]
This paper explores whether knowledge of the time-series properties of the premium
in the pricing of forward foreign exchange can be usefully exploited in forecasting
future spot exchange rates. Signal-extraction techniques, based on recursive application of the Kalman filter, are used to measure the premium. Predictions using premium models compare favourably with those obtained from the use of the forward
rate as a predictor of the future spot rate. The results also provide an interesting
description of the time-series properties of the premium
I. I N T R O D U C T I O N
Forward discount bias is a phenomenon thai has been
studied extensively in the literature. In addition to forward
exchange unbiasedness being rejected, it is generally found
that the change in the future exchange rate is negatively
related to the forward discount. A prominent explanation
for the rejection of forward rate unbiasedness is the existence of a time-varying risk premium. Other explanations
involve peso problems, irrationality of expectations, learning behaviour and market inefficiency. Useful surveys of
the empirical findings in this area are provided by
Hodrick (1987), Lewis (1995) and Engel (1996). In this
paper we will remain agnostic on the exact source of
forward exchange bias and refer to the 'premium', which
is not necessarily the same as a risk premium, but can also
capture elements of the other explanations. The same vocabulary is used in Fama (1984) and Clarida and Taylor
(1987).
There exists a substantial literature on the presence or
absence of premia (possibly time-varying) in the pricing of
forward contracts for foreign exchange. Conditional on the
hypothesis that the foreign exchange market is efficient or
rational, the existence of time-varying risk premia has been
documented in the literature, among others, by, Hansen
and Hodrick (1983), Fama (1984), Hodrick and
Srivastava (1984). Korajczyk (1985) and Wolff (1987a).
Deviations from rational expectations were documented.
among others, by Froot and Frankel (1989) and Cavaglia
et al. (1994).
In this paper we will explore whether knowledge about
the time series properties of premia can be usefully
exploited in forecasting future spot exchange rates. In
order to identify such premia. we follow Wolff (1987a)
and Nijman et al. (1993) in taking a signal-extraction
approach, based on recursive application of the Kalman
filter. This approach provides a natural framework for
modelling the premium as an unobservable component in
order to generate forecasts that take advantage of the time
series properties of the premium.
The paper is organized as follows. In Section II some
evidence on the properties of current forward and spot
exchange rates as predictors of future spot rates is presented. In Section III the methodology that is used in this
paper is explained; here the Kalman filter is described in
some detail. In Section IV the prediction results that were
obtain on the basis of the premium models that are constructed in the previous section are presented. Section V
offers some conclusions.
II. C U R R E N T SPOT A N D F O R W A R D R A T E S
AS P R E D I C T O R S O F F U T U R E S P O T
RATES
A number of researchers have investigated the properties of
forward exchange rates as predictors of future spot rates.
Applied Financial Economics ISSN 0960-3107 print/ISSN 1466-4305 online © 2000 Taylor & Francis Lid
h ttp: I/www .tandf,co,uk/journals
371
372
C C p. Wolff
Table 1. Summary statistics on the forecasting performance of
current forward and spot exchange rates as predictors of onemonth-ahead future spot i ates
Rate
ME"
MAE'
RMSE"
D'-Statistic
Eorward rate
$/pound
$/mark
$/yen
-0.22
-0.31
-0.15
1.98
2.38
2.13
2.57
3.13
2.94
.021
.017
.034
Spot rate
S/pound
$/mark
$/yen
-0.43
0.00
0.06
1.92
2.31
1.98
2.52
3,08
2.84
.000*
.000*
.000*
Note: "The Me, MAE, and RMSE are approximately in percentage terms.
'' The t/-statistic of the random walk forecast is one by definition
e.g. Hansen and Hodrick (1980). Bilson (1981), Meese and
Rogoff (1983), Fama (1984) and Wolff (1987b). The
general consensus is that forward rales are not very good
predictors of future spot rates. Both Meese and Rogoff
(1983) and Fama (1984) demonstrated that the current
spot rate is a somewhat better predictor of the future
spot rate than current forward rate.
Table I presents summary statistics on the one-monthahead forecasting performance of the forward rate and the
once lagged spot rate for three important exchange rates
involving the US dollar: the dollar/pound., dollar/mark and
dollar/yen exchange rates. For reasons of comparison, the
original dataset that was employed by Wolff (1987a) and
Nijman et al. is studied (1993) to develop the unobserved
components models. Spot exchange rates and 30-day forward rates are taken from the Harris Bank Database that is
supported by the Center for Studies in International
Finance at the University of Chicago. The rates are logs
of US dollars per unit of foreign currency. There are 148
observations covering the period 6 April 1973 to 13 July
1984.
Forecasting accuracy is measured by four summary statistics that are based on standard symmetric loss functions:
the mean error (ME), the mean absolute error (MAE), the
root mean squared error (RMSE) and the (.'-statistic.
Theirs ^'-statistic is the ratio of the RMSE to the RMSE
of the random walk forecast. Because we are looking at the
logarithm of the exchange rate, the ME, MAE and RMSE
are unit-free (they are approximately in percentage terms)
and comparable across currencies. By comparing predictors on the basis of their ability to predict the logarithm
of the spot exchange rate, we circumvent any problems
arising from Jensen's inequality. Because of Jensen's
inequality the best predictor of the level of the spot
exchange rate expressed as unit of currency / per unit of
currency / may not be the best predictor of the level of the
spot exchange rate expressed as units of currency / per unit
of currency /. Table 1 shows some basic forecasting results
Table : . Autocorrelations of F(t. t + 1) - S(t + 1)"
Lag
1
2
3
4
5
6
7
8
S/pound
S/mark
0.244**
0.14*
-0.02
-0.04
0.06
0.09
0.09
0.03
S/yen
0.15*
0 02
0.04
-0.11
-0.09
-0.08
0.02
0.12
0.19**
&.Mi
&M
^M
0.09
-0.02
-0.10 .
0.04
Note:" F{tJ-\- 1) is the log of the forward exchange
rate al time t for currency to be delivered at time
/ + I and 5{ / + 1) is the actual, realized value of the
spot rate at time / + 1.
** Significantly different from zero at the 5% level.
* Significantly different from zero at the 10% level.
for our dataset. Consistent with the previous literature the
results in Table 1 indicate that the current spot rate is a
better predictor of the future spot rate than the current
forward rate, both in terms of mean absolute error and
root mean squared error.
In Table 2 autocorrelations of the forecast error
F(/,? + 1) - 5 ( ? + 1 ) , the log of the thirty-day forward
rate, observed at month t for a contract maturing at
/ + ! , minus the log of the corresponding spot rate
observed around the maturity date are presented. Significant autocorrelations are reported in a number of cases,
in particular at lag one. These autocorrelations give us
information about the time series properties of the underlying premium terms. In Section HI a hypothesis concerning
the autocorrelation patterns that are observed in Table 2 is
advanced.
III. MODELLING THE PREMIUM
The model
The forward exchange rate can be conceptually divided
into an expected future spot rate and a premium term:
F{tj+\)
= E[S[t^\)\t\
+ P{t)
(1)
where E[s{t + \)\t\ is the rational or efficient forecast of the
log of the spot rate at time r + 1. based on all information
available at time /, and P{t) is the risk premium. Different
asset pricing models give different expressions for the premium term. Subtracting 6 ' ( / + l ) from both sides in
Equation 1. we obtain
,
f (r, t+\)-S{t
+ \)^
E[S{t + 1)] - 5(? + 1) + P[t)
(2)
and if we define v[t) = {E[S{t-\- 1)|/] - S(/ -(- 1), we have
F{tj+\)-S{t+\)
= P[t) + v{t)
(3)
where \v{t)] is an uncorrelated. zero mean sequence, given
our assumption of rational expectations. Equation 3 states
Foreign exchange rates
373
that the forecast error of the forward rate as a predictor of
the future spot rate consists of a premium component and
a white noise error term due to the arrival of new information concerning the spot rate between times / and / + I. If
we have information about the time series properties of the
premium term, this knowledge could potentially be usefully
combined with the forward rate in generating forecasts of
future spot exchange rates. Note that this methodology is
not necessarily restricted to identifying risk premia: if persistent peso problem terms, market inefficiencies, or terms
related to learning behaviour are present, they can be
•picked up' in the same manner.
From Equation 3 it follows that the autocorrelations
that were presented in Table 2 in the previous section are
also the autocorrelations of the combined time series process [P{t) +1'(0]- It is convenient to refer to the premium
component P{t) as the signal that we would like to characterize and to v{t) as noise that is added to the signal. The
problem that we face can thus be characterized as extracting a signal from a noisy environment.
In this section we take a Kalman-filtering approach to
signal-extraction. In order to apply the Kalman filter, some
assumptions about the time series properties of the premium must be made. On the basis of the autocorrelation
patterns that we reported in Table 2. we conjecture that the
premium process [P{t)] may be adequately described by an
autoregressive time series process of order one:
where a is a constant autocorrelation coefficient and [^{0]
is an uncorrelated, zero mean sequence. It can be shown
that, under this assumption together with the assumption
that \Q\ < 1. the combined process [/*(/) -I- ^{t)] has autocorrelations PI for j — 1,2,3,..., where pj is defined as:
Pj — a' — c/ / [[a} / o^,) / {{ — a") + 1]
f. [i'{t),^{t)] and P{r) are independently distributed for
all r, /.
With these assumptions, the system Equation 3 ^ is
easily recognized as a state-space model, which can be
recursively estimated by means of the Kalman filter (see
e.g. Anderson and Moore, 1979). Equation 3 is the observation equation and Equation 4 is the state-transition
equation.
The Kalman Filter
It is assumed that the error terms /'(/) and ^(/) are normally
distributed and that the premium P{t) has a prior distribution with mean P{0\0) and variance I](0|0). At every
point in time i after the history of the process
i?(/) ^ [F(r,r-|-1)-.v(r-l-1); r^l
/] was observed,
we want to revise our prior distribution of the unknown
state variable P{t). The Kalman Filter allows tis, given
knowledge of P(0\0), S(0|0) and the ratio aj/al, to
compute recursively the mean and variance of P for each
subsequent period.
Denote the conditional distribution of P{t) given R(t) by
p[P{t)\R{t)]. Given our normality assumptions./?[/*(/)|/?(/)]
and p[P{t + \)\R{t-\- 1)] are also normal and completely
characterized by their first two moments. If we denote
the mean and variance of p[P{t)\R{t)] by P(t\t) and (E/j/)
respectively, and those of P[P{t + \}\R{t)] by P{t + \\t)
and I1(/ + 1|A). then the Kalman Filter recursions for
/ — 0 , 1 , 2 , . . . , are given by Equations 6-10:
P{f-\-l\t)
= P{i\t
(6)
(7)
Pit+\\t+
l) = P{t+
\\t)+k(t+
(5)
Here a^ and a,, denote the variances of ^{t) and u{t), respectively. For instance, if a = 0.9 and a^/al = 0,05, then
we obtain p, = 0.188, P2 = 0.169, p, = 0.152, etc. Thus, we
expect to find fairly small positive autocorrelations that
decay exponentially from the starting value p[. Given the
fact that autocorrelations of such small magnitude are hard
to detect in finite samples, we argue that the results in
Table 2 are roughly consistent with our hypothesis concerning the time series properties of P{t).
In the forecasting experiment that is undertaken in the
next section, the two-equation mode! given by Equations 3
and 4 is considered. In addition, the following properties
IK' assumed:
a. E[v{t)] = 0 , vav[v{t)] = aj,
c. [vit]] is an independent sequence
d. [^(l)] is an independent sequence
e. vit) and ^{r) are independently distributed for ail r.t
P{t + \\t)]
(8)
(9)
with
(10)
Without the normality assumptions, the above results hold
for best linear unbiased prediction rather than conditional
expectations.
IV, PREDICTION RESULTS ON THE BASIS
OF PREMIUM MODELS
In this section the Kalman Filter is employed for recursive
signal-extraction and spot rate prediction. Predicted values
374
C C. P. Wolff
U-staUstic
for the spot exchange rate at time / + 1. given the information that is available at time /, can be generated as
- f ( ^ / + \)-aE[P{t-
\)\t]
(11)
Thus, the forward rate as a predictor of the future spot rate
is essentially 'corrected' for a premium term.
In order to be able to apply the Kalman Filter, two parameters need to be specified: the state transition parameter
a and the variance ratio (TC/CT;;. For convenience, we will
define A ^ oljo',,. Since information about the magnitudes
of a and A is difficult to obtain, a search procedure is
implemented: we search over a plane of values for the parameters a and log|o(A). Given a combination of values for
a and logm(A), predictions for future spot rates can be
generated from Equations 3 and 4 on a period by period
basis and summary statistics on the model's forecasting
performance ean be calculated. Values of o in the range
0.00-1.10 (grid size 0.01) and values of log|n(A) in the range
-tO.0-0.0 (grid size 0.1) are tried.
To he able to start off the Kalman filter we also need to
specify the starting values P(0|0) and E(0|0). In order to
obtain reasonable values for /'(0|0) and S(0|0). we
estimated Equation 12 by ordinary least squares (OLS):
/ • ( / , / - ! - I ) - 5 ( / - f 1) ^constant + error term
0.995 0.990
0.0
0.1
0.2 0.3
0.4 0.5
0.6
0.7 0.8
0.9
1.0
Fig. I. U-statistics on the forecasting performatice of the premium
model for the dollar jpound exchange rule for different values of the
stute transition parameter a. (logio(cr^/o-r/) = -1-42).
(12)
using the first 25 observations of our sample. The estimated
intercept and its estimated variance were then used as prior
mean and variance for the state variable P{t). The Kalman
filter was then started of for the 26th observation and run
for two periods before forecasts were generated. The model's ability to predict one-month-ahead spot exchange rates
is judged mainly on the basis of the root mean square
forecast error (RMSE). In both the a and A dimensions
distinct global minima are found in terms of the RMSE
criterion, for all three exchange rates.
In Figs 1-3 the model's forecasting performance for
different values of the autocorrelation coefficient a is
shown for the three exchange rates. C/-statistics are
reported in the figures, i.e. RMSEs are scaled by the
RMSEs of corresponding random walk forecasts. The Vstatistics in Figs 1-3 are calculated for the values of A for
which overall minimum RMSEs are attained. For comparison, levels of the (/-statistics that result from the use
of the forward rate and the lagged spot rate as predictors of
the four-week-ahead spot exchange rate are indicated in the
graphs.
Similarly, in Figs 4-6 the model's forecasting performance is shown for different values of A (for the values of a
for which overall minimum RMSEs are attained). Several
comments are in order with regard to these figures. First,
the premium models do better than the forward rate over
wide ranges of parameter values. Useful information concerning the future values of spot exchange rates is being
U-statisttc
0.04S 0.040 0.035 1.030
forward
rat«
1.025
1
-
1.02.1 -
-
^
preilua lodsl
1.015 1.010 -
\
j
\J
1.005 -
randoB walk
0.995
0.990
0.0
1
0.1
..1
1
0.2 0.3
1
0.4
t..
t
0.5 0.6
1
1
0.7
0.8
a
1
1..,,
0.9 1.0
Fig. 2. U-statistics on the forecasting performance of the premium
model for the dollar/mark exchange rate for different values of the
.state transition parameter a. {\o%^Q{(T\jai) = -2.38).
Foreign exchange rates
375
l>-SUtisUc
U-statlstlc
1.000
valk
0.995
0.990
0.0
•r
..)
I.
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Fig. 3. U-statistics on the forecasting performance of the premium
model for the dollarjyen exchange rate for different vahtes of the
stale transition parameter a. (log|g((Tf/a,,) = —0,94).
-6.4
-5.4
-4.4
-3.4
-2.4
-1.4
Fig. 5. U-statistics on the forecasting performance of the premium
model for the dollarlmark exchange rate for different values of
U-staUstic
U-statistic
1.050
-6.8
-5.8
-4.8
-3.8
-2.8
-i.e
-0.8
Fig, 4. U-statistics on the forecasting performance of the premium
model for the dollarlpound exchange rate for different values of
-3.9
-2.9
-1.9
-0.9
Fig. 6. U-statistics on ihe forecasting performance of the pretnium
model for the dollar/yen exchange rate for different values of
C. C. P. Wolff
376
Table 3. Pwametcr values of ftw premium models that
lead to )}iiiiinntni RAfSE onc-step-ahend forecasts
ami summary statistics on these models' forecasting
performance
$/pound
(I
A
SNR"
ME''
MAE''
RMSE''
(/-stat.
0.88
0.038
0.169
-0.10
2.02
2.68
1.003
$/mark
0.98
0.004
0.103
-0.24
2.22
3.03
I.OIO
$/yen
0.63
0.115
0.191
-0.17
2.39
3.20
1.008
Note: "SNR is signal-to-noi.se ratio
' Approximately in percentage terms.
picked up by our empirical characterization of the time
series properties of the risk premia. Second, the premium
models never do better than the random walk forecast:
(/-statistics smaller than one are not attained. Third, the
graphs all show distinct global minima in terms of the
RMSE criterion.
The locations of these minima provide us with information about the premium processes in the sense that the best
empirical description of the risk premia can be expected to
give the best empirical description of the premia can be
expected to give the best forecasting results. In Table 3
the parameter values of the premium models that lead to
minimum RMSE one-step-ahead forecasts are presented
for the three currencies. In the third row of the table we
report a signal-to-noise ratio (SNR): the ratio of the variance of the premium P{t) to the variance of the noise term
v{t) that is implied by the reported values of a and X. In the
case at hand the SNR is defined as ( l / I - a ~ ) A . In
the lower part of the table detailed summary statistics on
the forecasting performance of the minimum RMSE
premium models are presented.
Again, several comments are in order. First, the premium models that perform best in terms of RMSE of
four-week-ahead spot rate forecasts have fairly high, positive first-order autocorrelation coefficients (a's). Second.,
the signal-to-noise ratios are in the range 0.10-0.19. They
imply that roughly 9-16% of the variance in the forecast
errors F{t,l + I,/) — 5(/ + 1) is due to variation in the premium term /*(/). Third., the (/-statistics indicate that,
although the premium models do not outperform the random walk forecasting rule, it is a very close contest indeed.
Perhaps the forecast form the premium models and the
random walk model can be usefully combined along the
lines indicated by Granger and Newbold (1977) or Thisted
and Wecker (1981).
V. CONCLUSIONS
I
This paper investigated whether knowledge about premia
in forward exchange rates can be usefully exploited in order
to forecast future spot rates. On the basis of signalextraction models in which premia are assumed to follow
AR(1) processes, we find for a wide range of parameter
values that "correcting' the forward rate for a premium
term leads to one-step-ahead forecast errors that compare
favorably with those obtained from the forward rate
without correction. The premium models that are most
successful in predicting future spot rates imply that 916Vo of the variance in the forward forecast errors is due
to variation in the premia. However, in keeping with earlier
literature on the forecasting performance of exchange rate
models (see Meese and Rogoff. 1983 and Wolff. 1987b). the
models that we develop are unable to outpertbrm the
simply random walk forecasting rule in a prediction
experiment.
Finally, the methodology that we employ in this article is
fairly general in the sense that it can be straightforwardly
applied to other financial markets, such as futures markets
and markets for government debt instruments.
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