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6.6. NOISE-TRADERS 203Problems1. (Siegel’s [128] Paradox) Let S t be the spot dollar price of the euroand F t be the 1-period forward rate in dollars per euro. The claim isif investors are risk-neutral and the forward foreign exchange market ⇐(128)is efficient, the forward rate is the rational expectation of the futurespot rate. From the US perspective we write this asE t (S t+1 )=F t .The risk-neutral, rational-expectations, efficient market statement froman European perspective is(1/F t )=E t (1/S t+1 )since from the euro-price of the dollar is the reciprocal of the dollareurorate. Both statements cannot possibly be true. Why not? (Hint:Use Jensen’s inequality).2. Let the Euler equation for a domestic investor that speculates in forwardforeign exchange beF t = E t[u 0 (c t+1 )(S t+1 /P t+1 )]E t [u 0 ,(c t )/P t+1 ]where u 0 (c) is marginal utility of real consumption c and P is thedomestic price level. From the foreign perspective, the Euler equationis1= E t[u 0 (c ∗ t+1 /(S t+1Pt+1 ∗ )]F t E t [u 0 (c ∗ t )/P t+1 ∗ ]where c ∗ is foreign consumption and P ∗ is the foreign price level.Suppose further that both domestic and foreign agents are risk neutral.Show that Siegel’s paradox does not pose a problem now that payoffsarestatedinrealterms. 163. We saw that the slope coefficient in a regression of s t −s t−1 on f t −s t−1is negative. McCallum [103] shows regressing s t − s t−2 on f t − s t−2yields a slope coefficient near 1. How can you explain McCallum’sresult?16 Engel’s [43] empirical work showed that regression test results on forward exchangerate unbiasedness done with nominal exchange rates were robust to speciÞcationsin real terms so evidently Siegel’s paradox is not economically important.