Page 164 - DECO503_INTERNATIONAL_TRADE_AND_FINANCE_ENGLISH

International Trade and Finance

Notes Theoretical framework
The forward exchange rate f observed at time t for an exchange at t + 1 is the market determined
t
s
certainty equivalent of the future spot exchange rate t1 . One way to split this certainty equivalent
+
into an expected future spot rate and a premium is
ES
t
F = ( t1 ) + P t ... (1)
+
s
where F = ln f , S + t1 = ln t1 , and the expected future spot rate, ( t1 ) , is the rational or
ES
+
+
t
t
efficient forecast, conditional on all information available at t. Logs are used (a) to make the analysis
independent of whether exchange rates are expressed as units of currency i per unit of currency j or
units of j per unit of i, and (b) because some models for the premium can be stated in logs.
Eq. (1) is no more than a particular definition of the premium component of the forward rate. To give
the equation economic content, a model that describes the determination of P, is required. Examples
of such models are discussed later. For the statistical analysis of the premium and expected future
spot rate components of the forward rate, however, it suffices that the forward rate is the market
determined certainty equivalent of the future spot rate.
Statistics
From (1) the difference between the forward rate and the current spot rate is
F – S t = P 1 + E S +1 − S ) t . ... (2)
t
( t
F – S
F – S
S
+
t
Consider the regressions of t + t 1 and t1 − S (both observed at t + 1) on t t (observed at t),
F – S + t 1 = α + β F – S t ) ε+ 1,t +1 , ... (3)
1
1
( t
t
S + t1 − S = α + β 2 ( t t ) ε+ 2,t +1 . ... (4)
F – S
t
2
Estimates of (4) tell us whether the current forward-spot differential, F – S has power to predict the
t
t
future change in the spot rate, S + t1 − S Evidence that β is reliably non-zero means that the forward
t
2
rate observed at t has information about the spot rate to be observed at t + 1. Likewise, since F – S + t 1
t
ES
ES
is the premium P plus ( t1 ) − S + t1 , the random error of the rational forecast ( t1 ) , evidence
+
t
+
F – S
1
that β in (3) is reliably non-zero means that the premium component of t t has variation that
F – S
shows up reliably in t + t 1 .
With the assumption that the expected future spot rate in the forward rate is efficient or rational, the
regression coefficients in (3) and (4) are
cov F – S ,F – S ) t
( t
β 1 = + t 1 t ,
σ 2 ( t ) t
F – S
σ 2 ( ) +P cov P ,E S − S ))
( t
( t
= 2 2 t +1 t , ... (5)
σ ( ) σ+P t ( (E S + t 1 − S t )) + 2cov P ,E S +1 − S t ))
( t
( t
cov S − S ,F – S ) t
( t1
β 2 = + t t
σ 2 ( t ) t
F – S
σ 2 ( (ES − S )) + cov P ,ES − S ))
( t1
( t
= + t1 t + t ... (6)
σ 2 ( ) σ+P t 2 ( (E S + t 1 − S t )) + 2cov P ,E S +1 − S t ))
( t
( t
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