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174CHAPTER 6. FOREIGN EXCHANGE MARKET EFFICIENCYCovariance decomposition and Euler equations. We will use the propertythat the covariance between any two random variables X t+1 andY t+1 can be decomposed asCov t (X t+1 ,Y t+1 )=E t (X t+1 Y t+1 ) − E t (X t+1 )E t (Y t+1 ).For a particular deÞnition of X and Y , the theory, embodied in (6.11)restricts E t (X t+1 Y t+1 ) = 0. Using this restriction in the covariancedecomposition and rearranging givesThe Real Risk PremiumE t (Y t+1 )= −Cov t(X t+1 ,Y t+1 ). (6.12)E t (X t+1 )(111)⇒Set Y t+1 =(F t − S t+1 )/P t+1 and X t+1 = µ t+1 in (6.11) and use (6.12)to get" #Ft − S t+1E t = −Cov h³ ´ iFt−S t+1t ,µt+1P t. (6.13)P t+1 E t µ t+1The forward rate is in general not the rationally expected future spot inthe Lucas model. The expected forward contract payoff is proportionalto the conditional covariance between the payoff and the intertemporalmarginal rate of substitution. The factor of proportionality is−1/E t (µ t+1 ) which is the ex ante gross real interest rate multiplied by−1.h iHow do we make sense of (6.13)? Suppose that E Ft−S t+1t P t+1< 0.Then the covariance on the right side is positive. You expect to generateaproÞt by buying the foreign currency (euros) forward and resellingthem in the spot market at E t (S t+1 ). A corresponding strategy thatexploits the deviation from uncovered interest parity is to borrow thehome currency (dollars) and lend uncovered in the foreign currency(euros). The market pays a premium to those investors who are willingto hold euro-denominated assets. It follows that the euro must be therisky currency. If you are holding the euro forward, the high payoffstates occur when h iF t −S t+1P t+1is negative. By the covariance term in(6.13), these states are associated with low realizations of µ t+1 . But