Wolff, Christian C.P.

English

Research Papers in Economics

Journal of International Financial Markets,
Institutions and Money 10 (2000) 1–8
Measuring the forward foreign exchange risk
premium: multi-country evidence from
unobserved components models

Christian C.P. Wolff *
Limburg Institute of Financial Economics (LIFE), Maastricht Uni6ersity, PO Box 616,
6200MD Maastricht, The Netherlands
Centre for Economic Policy research (CEPR), London, UK
Received 22 November 1998; accepted 30 January 1999
Abstract
We investigate the nature of the foreign exchange risk premium for a wide range of
currencies, using unobserved components models with exactly matched spot and forward
exchange rate data. Signiﬁcant time-variation of the risk premium is documented for most
currencies. Our estimates indicate considerable persistence in the risk premium, and suggest
that the variability of the risk premium is quite low relative to the variability of the forward
forecast error. © 2000 Elsevier Science B.V. All rights reserved.
Keywords: Forward exchange bias; Risk premium persistence; Signal extraction
www.elsevier.com:locate:econbase
1. Introduction
Forward discount bias is a phenomenon that was studied extensively in the
literature. In addition to the 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
 The author is grateful to Franz C. Palm for helpful comments on an earlier version of this paper.
* Corresponding author. Tel.: 31-43-3883838; fax: 31-43-3258530.
E-mail address: c.wolff@berﬁn.unimaas.nl (C.C.P. Wolff)
1042-4431:00:$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.
PII: S1042-4431(99)00021-92 C.C.P. Wolff : Int. Fin. Markets, Inst. and Money 10 (2000) 1–8
involve peso problems, irrationality of expectations, learning behavior, and market
inefﬁciency. Useful surveys of the empirical ﬁndings in this area are provided by
Hodrick (1987), Lewis (1995), and Engel (1996). In this article we attempt to obtain
more information about the nature of the risk premium. One approach to model
the risk premium is the unobserved components (or signal extraction) methodology
that was introduced in this literature by Wolff (1987) and Nijman et al. (1993).
They showed how the risk premium can be interpreted as an unobserved compo-
nent, and how models of this type can be identiﬁed and estimated. The available
empirical evidence related to this approach is quite limited as only one relatively
small sample, containing three currencies relative to the US Dollar, was studied. In
addition to being limited, this sample is by now fairly dated. The primary objective
of this article is to further assess the relevance of the unobserved components
approach by studying a large, up-to-date dataset, covering 20 years of exchange
rate data and 15 different countries. Care is taken, contrary to the earlier studies,
to follow the sampling procedure of Bekaert and Hodrick (1993) to match spot and
forward data, in order to avoid the introduction of measurement error.
2. The risk premium as an unobserved component
The logarithm of the forward exchange rate can be divided into an expected
future spot rate component and a risk premium component:
Ft
t1Et[St1]Pt (1)
where Ft
t1 is the natural logarithm of the forward rate at time t for a contract
maturing at t1, E[St1] is the rational expectation, based on information
available at time t, of the log of the spot exchange rate at time t1, and Pt is the
risk premium. Subtracting St1 from both sides of Eq. (1) and deﬁning 6t1
E[St1]St1, we obtain
ytFt
t1St1Pt6t1 (2)
Eq. (2) states that yt, the forecast error resulting from the forward rate as a
predictor of the subsequent spot rate, consists of a risk premium and a white noise
error term: ‘signal’ and ‘noise’. We will attempt to model the signal as an
unobserved component in order to extract it from its noisy environment. An
important advantage of this methodology is that the expectation of the future spot
rate need not be modeled explicitly.
Our modeling strategy focuses on the premium itself. Following Wolff (1987),
Nijman et al. (1993) and Huisman et al. (1998), the signal is assumed to be
generated by a model from the autoregressive integrated moving average (or
ARIMA) class of models. This assumption is consistent with theoretical models
that have been studied in the literature, see Nijman et al., 1993. In order to explain
our modeling strategy, let us assume that Pt is generated by an ARMA(1,1) model:
(1fL)Pt(1vL)at (3)3 C.C.P. Wolff : Int. Fin. Markets, Inst. and Money 10 (2000) 1–8
where L is the lag operator, fB1, v01, f"v, and at is a white noise sequence
with mean zero and constance variance sa
2 which is uncorrelated with 6t% for all t and
t%. We assume that E[at6ti]0(Öi)
1. Substituting Eq. (3) into Eq. (2) we obtain.
(1fL)yt(1vL)at(1fL)6t1 (4)
The right-hand side of Eq. (4) consists of two uncorrelated MA(1) sequences. The
summation theorem for moving averages in Ansley et al. (1977) states that the
summation of two moving average processes of orders q1 and q2 has a MA(q*)
representation with q*0max[q1,q2]. Making use of this theorem, we can conclude
that yt in equation Eq. (4) has an ARMA(1,1) representation, so that Eq. (4) can
be rewritten as
(1fL)yt(1uL)ot (5)
where u01 and ot has mean zero and constant variance so
2.I fPt is generated by
a general ARMA(p,q) process, it can be shown that yt follows an ARMA (0p,
0max[p,q]) process, the parameters of which are identiﬁed as long as p]q1, see
Hotta (1989).
3. Data
In this study we employ end-of-the-month spot and forward exchange rates which
cover the period January 1976 through March 1996.
2 The maturity of the forward
rates is one month. All raw data are London closing mid-prices against the Pound
Sterling, obtained from Datastream, for 15 countries: Austria, Belgium, Canada,
Denmark, France, Germany, Ireland, Italy, Japan, the Netherlands, Norway, Spain,
Sweden, Switzerland and United States.
3 In our empirical analysis below all
exchange rates were crossed in order to report our results in US Dollar terms. The
data are sampled following the procedure described in Bekaert and Hodrick (1993),
using exact delivery dates of the forward exchange contracts. To ﬁnd the delivery
date on a forward contract made today, one ﬁrst ﬁnds today’s spot value date,
which is two business days in the future. Delivery takes place on the calendar day
in the next month that corresponds to the current spot value date, under the
condition that the delivery day is a business day. If not, delivery takes place on the
next business day if it falls within the same calendar month. If the latter condition
is not met, delivery takes place on the ﬁrst previous business day. This rule is
followed except when the spot value day is the last business day of the
1 This assumption could be relaxed in favor of allowing for (intertemporal) correlation between
forecast errors and risk premiums.
2 We thank Ronald Huisman for kindly providing the data employed in this study.
3 Note that the data for the case of Japan are only available from June 1978.4
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Table 1
Least squares estimates of ARMA model (5) for yt
f
a u
a Currency so
2 (10
4) sa
2 (10
4) sv
2 (10
4) Lower bound
b sa
2 (10
4) Upper bound
c sv
2 (10
4)
0.74*** 0.59*** 12.14 Austria 1.39 9.67 0.34 10.13
(0.16) (0.20)
0.80*** 0.68*** 12.67 Belgium 0.89 10.72 0.22 11.00
(0.15) (0.18)
– 0.09 1.98 Canada – – 0.18 1.63
(0.06)
0.77*** 0.63*** 11.81 Denmark 1.09 9.67 0.27 10.02
(0.15) (0.18)
France 0.81*** 0.69*** 11.65 0.72 9.99 0.18 10.21
(0.14) (0.18)
Germany 0.77*** 0.64*** 12.49 1.03 10.46 0.25 10.79
(0.16) (0.20)
0.74*** 0.60*** 11.92 Ireland 1.31 9.56 0.32 10.00
(0.17) (0.20)
– Italy 0.17*** 11.23 – – 1.96 7.64
(0.06)
– 0.20*** 14.26 Japan – – 2.82 9.18
(0.07)
0.72*** 0.56*** 12.33 Netherlands 1.61 9.64 0.39 10.18
(0.18) (0.20)
0.64*** 0.50* 9.10 Norway 1.34 7.12 0.32 7.61
(0.24) (0.27)
– 0.13** 11.90 – – 1.58 8.95 Spain
(0.06)
– 0.15** 10.45 Sweden – – 1.54 7.60
(0.06)
– 0.18*** 16.15 – – 2.92 10.84 Switzerland
(0.06)
– 0.13** 13.42 – – 1.79 10.09 UK
(0.06)
a Large-sample standard errors are provided in parentheses.
b The corresponding upper bound is so
2.
c The corresponding lower bound is zero.
* Denotes statistical signiﬁcance at the 10% level.
** Denotes statistical signiﬁcance at the 5% level.
*** Denotes statistical signiﬁcance at the 1% level.5 C.C.P. Wolff : Int. Fin. Markets, Inst. and Money 10 (2000) 1–8
current month, in which case the forward value date is the last business day of the
next month. Unless one follows these rules precisely, measurement error is intro-
duced into the analysis.
4. The empirical results
As we aim to estimate time-series models along the line of Eq. (5) for all available
currencies relative to the US Dollar, models need to be identiﬁed (in the Box-
Jenkins sense) for all of these currencies. This was accomplished using the proce-
dures described in Box and Jenkins (1976). Additional guidance was obtained from
the Schwarz information criterion for laglength selection. In all cases an
ARMA(1,1) or a MA(1) model for yt was found to be parsimonious and adequate.
Note that an ARMA(1,1) model for yt is consistent with an AR(1) or ARMA(1,1)
model for the risk premium, Pt, itself. Similarly a MA(1) model for yt is consistent
with a MA(1) model for Pt. Our estimated models for yt, therefore, at the same
provide information about the time-series behavior of the risk premium, Pt. Least
squares estimates for the models are presented in Table 1. In all cases a very
insigniﬁcant constant term was removed prior to estimation.
In the second and third columns of Table 1 the parameter estimates for the
ARMA(1,1) and MA(1) models, respectively, for yt are presented. Large-sample
standard errors are provided in parentheses. For Austria, Belgium, Denmark,
France, Germany, Ireland, the Netherlands and Norway we ﬁt ARMA(1,1) models
for yt, which are consistent with either an AR(1) or an ARMA(1,1) model for the
risk premium. All parameter estimates in these cases are highly statistically signiﬁ-
cant. If one is willing to restrict v a priori to be equal to zero, i.e. to choose the
AR(1) model for the risk premium — both sa
2 (the variance of the random
disturbance term in Eq. (3) for the risk premium) and sv
2 (the variance of the ‘noise’
Table 2
Tests of H0: fu for estimated ARMA(1,1) models
P-value Currency F-statistica
6.628** 0.0106 Austria
Belgium 0.0250 5.091**
6.182** 0.0136 Denmark
0.0319 4.659** France
0.0267 4.974** Germany
0.0117 6.447** Ireland
0.0087 7.007*** Netherlands
0.0234 Norway 5.206**
a Under the null hypothesis the test statistic is distributed as F(1,239).
** Indicates statistical signiﬁcance at the 5% level.
*** Indicates statistical signiﬁcance at the 1% level.6 C.C.P. Wolff : Int. Fin. Markets, Inst. and Money 10 (2000) 1–8
term in Eq. (2) above) can be identiﬁed from f, u, and so
2 (the variance of the
disturbance term in Eq. (5) for yt). The estimates of so
2, sa
2, and sv
2 (the latter two
variances under the assumption that v0) are reported in columns four through
six. If one is not willing to restrict the risk premium to being generated by an AR(1)
model, but allows it to be governed by an ARMA(1,1) model, sa
2 and sv
2 are not
identiﬁed. Nijman et al. (1993), however, derived bounds on these two variances:
so
2[(1u
2)f(1f
2)u]:(1f)
20sa
20so
2 and 00sv
20so
2(1u)
2:(1f)
2. These
bounds are presented in the seventh and eighth columns in the table.
For Canada, Italy, Japan, Spain, Sweden, Switzerland and the United Kingdom
we ﬁtted MA(1) models. These are consistent with MA(1) models for the underlying
risk premium processes, too. The estimates for the MA(1) coefﬁcient u are reported
in the third column. For these models the underlying parameters v, sa
2, and sv
2 are
not identiﬁed, but bounds on the variances are again available: uso
20sa
20so
2 and
00sv
20(1u)
2so
2. For the MA(1) models these bounds are reported in the seventh
and eighth columns. Note that all estimated u-coefﬁcients are highly statistically
signiﬁcant, with the exception of the Canadian Dollar.
For all of the models (except in the Canadian case) there is evidence of
nonnegligible time variation in the risk premium. It is interesting to note that the
models’ point estimates for the parameters f and u imply that yt displays positive
ﬁrst-order serial correlation. For instance, the point estimate u0.20 for the
Japanese Yen implies a ﬁrst-order autocorrelation for yt of u:(1u
2)0.19, see
Box and Jenkins (1976). This positive serial correlation was of course also apparent
at the identiﬁcation stage of the modeling procedure. The case of the Canadian
Dollar is quite different from the other currencies in terms of statistical signiﬁcance.
This is likely to be related to the fact that the variance of the forward forecast error,
the dependent variable, for the Canadian Dollar is only about ﬁfteen percent of the
average variance level for the other currencies. For the ARMA(1,1) cases, if again
one is willing to assume that v0, the positive f-estimates would correspond
directly to estimated AR(1) coefﬁcients for the underlying risk premium models.
These estimates would thus indicate considerable persistence in the risk premium.
For the MA(1) models it can be shown that the negative u-estimates must also
imply negative v-values, which in turn implies, again, that the risk premia show
persistence (in the sense of positive serial correlation) over time.
When considering the point estimates of f and u for the ARMA(1,1) models, one
notices that they are of similar orders of magnitude in all cases. This issue merits
investigation because if indeed they are identical to each other, (1-fL) will cancel
(1-uL), so that the forward forecast error would be a white noise process. Formal
(large-sample) tests of H0: fu are provided in Table 2.
Table 2 provides F-statistics and associated P-values. The results are quite
uniform: in all cases equality of f and u is rejected at the ninety-ﬁve percent
conﬁdence level and sometimes (the case of the Netherlands) even at the ninety-nine
percent level. Hence, we can conclude that indeed the persistence described above
is present also in the ARMA(1,1) cases.
It is also interesting to study the various variance estimates in detail. First, we
consider the ARMA(1,1) models. Here, so
2, the variance of the innovation in the yt7 C.C.P. Wolff : Int. Fin. Markets, Inst. and Money 10 (2000) 1–8
process is a useful benchmark. If we are willing to assume that v0, the
relatively low sa
2 values indicate that the variability of the risk premium is quite
low relative to the variability of the forward forecast error, as well as relative to
the ‘noise’ variance sv
2, which is a conclusion that differs from the earlier result
in Wolff (1987). If we are not willing to assume that v0, the variance bounds
in the last two columns are appropriate. They are quite wide and, therefore, not
very informative. The variance bounds for the MA(1) models are somewhat more
informative, but still do not allow for precise comparisons of variance magni-
tudes.
5. Conclusions
On the basis of unobserved components models we have studied the nature of
the foreign exchange risk premium in this article for ﬁfteen currencies relative to
the US Dollar. For all of the currencies but one, we ﬁnd signiﬁcant evidence of a
time-varying risk premium. The stochastic structure of the forward forecast error
can be captured adequately with low-order models of the (AR)MA class. Our
model estimates suggest persistence in the risk premium, in the sense of positive
serial correlation, thus corroborating the results of Nijman et al. (1993) and
Wolff (1987). For a number of models our results indicate (assuming an AR(1)
model for the risk premium) that the variance of the risk premium is quite low
relative to the variance of the forward forecast error. It is interesting to note that
we have identiﬁed two different stochastic structures for our samples: an
ARMA(1,1) model applies to a subset of continental European countries (includ-
ing ‘core EMS’ countries). The model estimates are quite similar for all these
cases. An MA(1) model applies to almost all of the remaining countries (Canada
being the exception). Within this group the model estimates are also very similar
for different currencies.
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Common factors and tests for the unbiasedness hypothesis.

Forward foreign exchange rates, expected spot rates, and premia: a signal-extraction approach.

Identiﬁcation of unobserved components models.

On biases in the measurement of foreign exchange risk premiums.

On the structure of moving average processes.

Premia in forward foreign exchange as unobserved components.

The empirical evidence on the efﬁciency of forward and futures foreign exchange markets.

The forward discount anomaly and the risk premium: a survey of recent evidence.

Time Series Analysis, Forecasting and Control, revised edition. Holden-Day,

Measuring the forward foreign exchange risk premium: multi-country evidence from unobserved components models.

http://arno.unimaas.nl/show.cgi?fid=10646

Measuring the forward foreign exchange risk premium: multi-country evidence from unobserved components models.

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