Barton, N.

Yiu, M.

English

Mathematics in Industry Information Service Eprints Archive

LONG TERM FOREIGN CURRENCY EXCHANGE RATE
PREDICTIONS
The motivation of this work is to predict foreign currency exchange rates
between countries using the long term economic performance of the respective
economies. The predictions are long term - approximately one year in advance
- and it is recognised that short term predictions (on a daily basis or shorter)
require different approaches to those presented here.
The problem was presented to the Study Group by World Values Ltd. This
company has provided advice for many years on foreign currency exchange rates
using a model entitled "the currency evaluator", the derivation of which reflects
the extensive experience of the principals of the company. In brief, points are
awarded to each country in respect of its economic performance in eight key
economic properties. The points are summed for each country over all the prop-
erties, and a comparative ratings or performance table is constructed. Minor
adjustments are made if necessary, to this table in the light of practical ex-
perience - for example, if a country always scores well with respect to certain
economic properties, then the effect of those properties might be discounted
somewhat in predictions. The ratings table is then used to make predictions for
exchange rates one year ahead. Some details of this process have been published
by World Values Ltd (1989).
The World Values model is not optimized in the usual mathematical sense
in which parameters are determined by fitting predictions to data. Rather, the
model has been tuned according to practical experience. The World Values
model is successful, but has room for improvement. The opinion of the dele-
gates at the Study Group was that it was more important for them to suggest
alternative mathematically rigorous frameworks for foreign currency predictions
than to suggest incremental improvements to the World Values model.
The eight key economic properties used by World Values Ltd to make long
term foreign currency exchange rate predictions are described in Section 2. These
properties have been identified through years of experience, rather than through
rigorous economic modelling.
Prior to the Study Group, a small amount of computer modelling was un-
dertaken in an attempt to improve the World Values forecasts. The results
of this exercise, incorporating some suggestions made at the Study Group, are
presented in Section 3. Under specific criteria which are described later, the
preliminary results are superior to those achieved by the World Values model.
Our preliminary model possessed some shortcomings from the point of view
of computational efficiency. Accordingly an improved model was suggested at
the Study Group and is described in Section 4. This model is linear and is
appropriate when we anticipate that the economic performance of a country
has a more or less constant effect on the exchange rate. The parameters in the
model are the same for all countries at all times. In our opinion, this linear
model should be constructed first in any subsequent computer modelling.
In fact, however, it is reasonable to expect that economic performance over
time has a variable effect on exchange rates. That is, stochastic properties should
be considered. Accordingly, a more general model incorporating variability was
developed at the Study Group and is described in Section 5. This model is based
on the Kalman filtering approach and would be more elaborate to implement
than the linear model described in Section 4.
The contentious issue of the short term effect versus long term effect of inter-
est rates was discussed at the Study Group. These discussions were inconclusive.
For completeness, however, we have included a discussion on this topic in Section
6. The discussion incorporates a practical suggestion to explore the short term
importance of interest rates.
All models for foreign currency exchange rate prediction rely on a large num-
ber of properties and variables. The following notation has been adopted:
m, n refer to countries (m, n = 1, ,8)
z refers to time periods (i = 1, , I).
In general, we have used quarterly data
and forecasts.
J refers to economic properties (j = 1, ... , J)
p:nn predicted value of 1 unit of currency m
in terms of currency n at time ti
P,~m log P~n
q:nn actual (market) value of P~n
Q~m log q~n
economic property j of country m
at time ti
dimensionless form of e~j
_l_(ei*. _ e .i*)
"," mJ J
J
These scaled (dimensionless) properties have the ad'vantage that e~j
e~l % increase p.a. of c.p.i.
e~2 % increase p.a. of real money (M3)
e~3 long term interest rates
e~4 % increase p.a. of average wages
e~5 current account balance as % of GDP
e~6 = trade balance as % of imports
e~7 financial reserves: the number of months
of last months imports that financial reserves will buye:~8 = % increase p.a. of G.N.P.
Further descriptions of these eight properties are given by World Values
Ltd (1989); and \ve note that preliminary smoothing procedures are sometimes
applied before the properties are used in forecasts. For use in Section 6, we
define
e~9 short term interest rates
(e.g. 90 day bank bill).
\Ve also use the concept of 'value' of currencies, but only in a relative sense.
Let
For our preliminary model, Wehypothesize that the present value of a cur-
rency is related linearly to the eight dimensionless economic properties four
quarters earlier:
8
vi+4 = vi {I + ~ a -ei .}m m ~ J mJ
j=l
The forecast period of four quarters is the same as in the World Values model.
This leads to the predicted exchange rates
In this model, the linearity inherent in the hypothesis (3) has been lost. More-
over, there are a number of parameters to be determined, namely eight coef-
ficients aj (j = 1, ,8) and seven initial exchange rates P~n(m fixed,n =
1, ... , m - 1,m + 1, 8). From these seven initial values, all other initial values
can be determined using identities (1,2).
Minor modifications were made to the model to predict P~n' rmn and P~n,
given initial values P~m' These modifications took the form (for k= 1,2,3)
The choice of weights in the least squares expression (6) should depend on the
estimate of P~n rather than qtnn because the presence of observational errors
inside the weights leads to biased estimates and inflated variances. The choice
of qtnn was made out of convenience, bearing in mind the time constraints of
the IvIISG. In any subsequent investigation, the weights must be carefully cho-
sen. The minimization algorithm used was the LMM (Levenberg, Marquadt and
Morrison) algorithm for non-linear least squares. In this algorithm, derivatives
of P~n with respect to the parameters are required, although these details are
not included here.
To use the preliminary model for predictions, results available at time tf were
used to predict values at t1+4' This procedure was repeated for I = 6, ... ,13.
~
~
1987 1968
'0 10
0
~
~
1987 1988
'0 10
N
1988 1988 1988 1989
20 30 40 1Q
Currency 3 against currency 6
1988 1988 1988 1989 1989
20 JO 40 1Q 2Q
Currency 3 against currency 7
1988 1988 1988 1988 1989 1989 1989
1Q 2Q 30 40 1Q 20 30
Currency 2 against currency 6
Figure 1: Illustra.tive one year forecasts of foreign currency exchange rates com-
pared with actual values. Legend: 1 - actual exchange rates; 2 - World Values
Ltd forecast; 3 - method of Section 3.
Figure 1 shows sample predictions of the preliminary model compared with
actual exchange rates and the predictions of the World Values model. Objective
comparisons between competing models were made using the mean square errors
10 1 12
£mn = 8" L.J (Pmn - qmn)
1=10
which are tabulated below. In terms of this criterion, the predictions made using
our preliminary model were superior to the World Values predictions on 25 out
of the 28 cross rates.
rnn 2 3 4 5 6 7 8
1 0.24638 0.03613 0.11343 0.06963 1021.53 0.03788 1.00271
2 0.00104 0.00971 0.08369 174.63 0.10041 0.68421
3 0.03847 0.25675 358.43 0.36115 1.89157
4 0.06042 149.37 0.08583 0.40526
5 55.31 0.00474 0.02421
6 0.000004 0.000009
7 0.29705
Table 1: Mean square error £mn of the predicted foreign currency exchange
rate by the World Values model.
rnn 2 3 4 5 6 7 8
1 0.04516 0.00331 0.00624 0.03367 109.88 0.03097 0.38264
2 0.00255 0.01178 0.06695 159.80 0.05990 0.66391
3 0.03896 0.06645 76.71 0.07804 0.50907
4 0.03772 98.68 0.03571 0.37955
5 29.42 0.00058 0.00635
6 0.000001 0.000009
7 0.04188
Table 2: Mean square error £mn of the predicted foreign currency exchange
rate by the model described in Section 3.
Certain shortcomings of the preliminary model must be pointed out however.
The model provides estimates for the parameters aj and the initial values p?nn
at each time t1(I = 6, ... ,13). These parameters do not always converge with
increasing I, as is shown, for example, in the results in figure 2 for a8. Besides the
choice of weights in the least squares expression (6), there is another factor which
leads to biased estimates and inflated variances. Since the index set AI includes
cross-rates for currencies, there is the possibility of correlations in residuals if
exchange rates do not reflect market values, for example, through government
manipulation of certain currencies.
-0.004
-0.006
-0.008
-0.01
The model described in the previous section has the limitations that param-
eters are determined by minimizing a least squares expression in which parame-
ters occur non-linearly. Minimization therefore required the use of the powerful
package LMM. In the present Section, an alternative version of the model is de-
veloped in which parameters enter linearly in the least squares problem, thereby
permitting the use of simpler packages.
If we define p:nn = log p:nn and take logarithms in equation (2), we obtain
1'2 1'3
log(1 + r) = l' - "2 + "3 - ...
and retain only the first terms in log{1 + 2:J=1 Gje:n). This gives
8
Pi+4 _ pi + ""' .( i i )mn - mn L.J GJ emj - enj ,
j=1
which is an equivalent linear model to that in Section 3, but far simpler to
implement since the parameters Gj (j = 1, ... ,8) and the initial values P;;m
enter linearly into p:nn in the least squares expression
E= L
in which Q:nn = logq:nn. (Note that minor modifications as in equation (5) are
required to predict P~n, P;'n, P;;'n.) Determination of the parameters Qj and the
initial values P;;m requires the solution of a set of simultaneous linear equations
for which simple standard packages are available.
• Use results available at time tI to predict values at time tI+4 (I
6, ... ,13)
• Compare the predictions with those of World Values Ltd and those of the
preliminary model (Section 3)
• Investigate goodness of fit
• Investigate the behaviour of the parameters Qj and P:?m with respect to
sign, statistical significance (through the t test), and variability as I is
increased
It would be easier to undertake these investigations with the improved linear
model than with the preliminary model (Section 3). The improved model will
still lead to biased estimates for the reasons given in Section 3.
If the results of the investigation foreshadowed in the previous section showed
that the parameters in the model appeared to change with time, then this in-
dicates that a model which incorporates stochastic variability should be used.
Such a model is provided by a Kalman filtering approach (Anderson & Moore,
1979).
We start by rewriting equation (8) in the form
8
Pi+4 _ pi + '" i( i i )+ Ximn - mn ~ Qj emj - enj mn
j=1
where the parameters Q~ now vary with time and X:nn is assumed to be Gaussian
white noise with zero mean and variance a;.
In order to use the standard Kalman filtering algorithm, we write for the
state vector,
[pi pi-l pi-2 pi-3 ,i i-I i-2 i-3mn mn mn mn °1 Q1 Q1 Q1
A B1 B2 Bs
o A 0 0
E'= 0 0 A 0
0 0 0 A
where A is the 4 X 4 matrix
A~[~0 0 1 ]0 0 01 0 0
0 1 0
and ilj is the 4 X 4 matrix
X~n [X:;n30 0 0 ~I-300 0 ~~-30 00
... ~~-30 00]
and the ~;-3are Gaussian white noise with mean zero and variance at, so the
covariance matrix Bi of X~n is
02 0 0 0x
0 02 0 06
0 0 02 06
0 0 0 02
~8
The observation or measurement equation is
Q~n = IlV/nn + Z:nn
The Kalman filtering algorithm proceeds as follows: start with the initial
estimate
[p3 p2 pI pO 3 0mn mn mn mn 0'1 ... 0'1 ...
• •• O'~ ••• O'~]
and the properties a;, a€j' j = 1, ... ,8, a1mn and the covariance matrix R::nn
(of V~n). The prediction stage is
which yields V~~lli. The prediction of the observation vector is Q~~li
Q"i+lli = H Vi+1li + E(Zi+lli)mn mn mn·
Then calculate the covariance matrix R~~li (of V~~lli )
This prediction-correction procedure is to be repeated for all m, n in the index
set M.
• This model is in a standard framework, and convergence would be guar-
anteed if F had constant coefficients which is, in fact, not the case here
• If the behaviour of the parameters O'~ varies with time, then the Kalman
filtering approach might give superior performance to the improved linear
model of Section 4
• Comparisons of forecasts could be made, as before, through the mean
square error crnn
• The performance of the Kalman filtering algorithm depends on a;, a~j'j =
1, ... ,8, a2Z • These are "tuning" parameters and need to be chosen care-
mn
fully. Literature to guide in the selection of these parameters is available
(Anderson & Moore, 1979, Harvey & Peters, 1984) although expert as-
sistance would be required by World Values Ltd if this model were to be
implemented
• This model could also be used to investigate sensitivity to variations in the
forecast period
• The model developed in this Section does not, however, take account of the
constraints (1,2). To do this, the following straightforward modification is
made. Rewrite equation (12) as
Qi = HVi + Zi
i i i i i iTQ = (Q12"'" Q18, Q23' ... , Q28' ... , Q78)
H = hmn is a 28 X 64 matrix with non-zero entries as follows:
8:S m :S 13: hmn = 1 (n = 5), hmn = -1 (n = 4(m - 7) + 5)
14:S m :S 18: hmn = 1 (n = 9), hmn = -1 (n = 4(m - 13) + 9)
19:5 m:S 22: hmn = 1 (n = 13), hmn = -1 (n = 4(m - 18) + 13)
23 :S m :S 25: hmn = 1 (n = 17), hmn = -1 (n = 4(m - 22) + 17)
26:S m:S 27: hmn = 1 (n = 21), hmn = -1 (n = 4(m - 25) + 21)
m = 28: hmn = 1 (n = 25), hmn = -1 (n = 29)
[ i i-I i-2 i-3 i i-I i-2 i-3VI VI VI VI ... V8 V8 V8 V8
i i-I i-2 i-3 i i-I i-2 i-3]T
0'1 0'1 0'1 0'1 .. ·0'8 0'8 0'8 0'8
Zi - [Zi Zi Zi Zi Zi )T- 12' ... , 18' 23"'" 28"'" 78
The covariance matrix of the measurement error vector Zi can be written
as a2lI lIT if we assume that all the zfnn are Gaussian distributed with
zero mean and same variance a2• The modification of equation (11) is
again very straightforward.
• A special case is that o::n does not depend on t. If so, the model requires
less computational effort.
Foreign currency exchange rates are clearly affected in the short term by
interest rate movements. Such short term movements are additional to the long
term effects of interest rates. One approach was to try to identify the short term
interest rate component f in the expression
where p:nn is a predicted exchange rate, q:nn is the market exchange rate, and
the argument of the factor f includes variables that have to be identified. In
an attempt to specify the factor fmn, we examined the 'gain' g:n on investments
over a period ~:
Here v~ denotes the value of currency 171, and e~9 is the short term interest rate
at time ti for country m.
. .. . vi+Cl. vi . vi+Cl.
gt _ gt = vt [{(I + et* )~ _ ~} _ {(I + et* )_n_. - I}]
m n n m9 vt t n9 vtn vn n
In this expression, (v:;tCl./v~) and (v~+Cl./ v~) would be predicted by an appro-
priate forecasting model, whilst (v:n/v~) = q:nn is the market realization of the
exchange rate.
We looked for a general "principle of equilibrium" in which q~n adjusts
immediately to make g~ - g~ = O. If such a principle held, this would provide
a means for determining the factor fmn described above.
• what forecasting model to choose?
• what method to choose to give e~9? e.g. actual rates at ti, average rates,
expected rates, etc.
• how to relate this general principle of equilibrium to definite accounting
and economic identities? e.g. the current account must balance
Accordingly, we looked for a simpler approach and now recommend that
the dimensionless short term interest rate e~9 should be included in the list of
economic properties that affect long term foreign currency exchange rates. Minor
modifications would be required to incorporate e~9 in the models presented in
Sections 4 and 5. Standard statistical tests could then be used to decide whether
or not to include e:n9' In particular, the Kalman filtering approach would lend
itself well to incorporation of e~9 in the correction (and not prediction) stage.
The preliminary model described in Section 3 yields promising results. Fur-
ther development of this preliminary model is possible, but our opinion is that
further development would be expedited using the improved linear model de-
scribed in Section 4. This follows since the improved linear model has a simpler
theoretical framework and would involve less computational work. The Kalman
filtering model would be more difficult to use than the model of Section 4, but
it would be necessary if the parameters O'j exhibited variability with time. In
cases where such variability occurs, the Kalman filtering model is optimal.
The models described in this report should have the capability to improve
the foreign currency exchange rate forecasts made by \Vorld Values Ltd.
The moderators for this problem (Noel Barton and Matthew Yiu) would like to
acknowledge the support and contributions of Mr James Cowan (vVorld Values
Ltd). The following are also to be thanked for their substantial contributions
to discussions on this problem: Basil Benjamin, Murray Cameron, Margaret
Gutowski, Joseph Ha, Peter Price, Nancy Spencer and Steve Spencer.
B.D.a. Anderson & J.B. Moore, Optimal Filtering (Prentice-Hall, Englewood
Cliffs, New Jersey, 1979).
A.C. Harvey and S. Peters, "Estimation procedures for structural time-series
models" London School of Economics, Discussion Paper No. 28 (1984).


Long term foreign currency exchange rate predictions

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