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200CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYUsing the method of undetermined coefficients, you can verify thatis a solution.∆s t = 1 ρ x t −(1 − µ)n t − (1 − µ)u t−1 , (6.66)ρProperties of the SolutionFirst, fundamentalists and noise traders both believe, ex ante, thatthey will earn positive proÞts from their portfolio investments. It is thedifferences in their beliefs that lead them to take opposite sides of thetransaction. When noise traders are excessively pessimistic and takeshort positions in the dollar, fundamentalists take the offsetting longposition. In equilibrium, the expected payoff of fundamentalists andnoise-traders are respectivelyE t ∆s t+1 − x t = −(1 − µ)n t , (6.67)E t ∆s t+1 − x t = µn t . (6.68)On average, the forward premium is the subjective predictor of thefuture depreciation: µE t ∆s t+1 +(1−µ)E t ∆s t+1 = x t . As the measure ofnoise traders approaches 0 (µ → 1), the fundamentals solution with notrading is restored. Foreign exchange risk, excess currency movements,and trading volume are induced entirely by noise traders. Neither typeof trader is guaranteed to earn proÞts or losses, however. The ex postproÞt depends on the sign of∆s t+1 − x t = −(1 − µ)n t + 1 ρ [1 − k(1 − µ)]v t+1 − 1 − µρ u t+1, (6.69)which can be positive or negative.Matching Fama’s regressions. To generate a negative forward premiumbias, substitute (6.62) and (6.52) into (6.66) to get∆s t+1 =[1− k(1 − µ)]x t + ξ t+1 , (6.70)where ξ t+1 ≡ (1/ρ)[1 − k(1 − µ)]v t+1 − (1 − µ)/ρu t+1 − (1 − µ)u t isan error term which is orthogonal to x t . If [1 − k(1 − µ)] < 0, the