Why is it true that the hypothesis that the forward exchange rate is an unbiased predictor of the future spot exchange rate is equivalent to the hypothesis that the forward premium
(or discount) on a foreign currency is an unbiased predictor of the rate of its appreciation (or depreciation)?
Answer: When the forward exchange rate is an unbiased predictor of the future spot exchange rate, we know that the forward rate equals the conditional expectation of the future spot rate. For example, at the 90 day maturity, we have
F(t,90) = Et [ S(t + 90) ]
Because the current spot rate, S(t), is in the information set that is used to take the conditional expectation, we can divide by it on both sides of the above equation. Subtracting one from both sides then gives
[ (F(t,90) - S(t)) / (S(t)) ] = Et [ (S(t + 90) - S(t)) / (S(t)) ]
This equation states that the forward premium on the foreign currency equals the expected rate of appreciation of the foreign currency.
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