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Looking at the MSCI data I don't see an effect of the dollar exchange rate. The fact that daily and monthly returns are in opposite directions is commonplace, even without currency conversions. I suspect that any discrepancies you're seeing between various EAFE indices and the final I fund value are due to timing differences - i.e. exactly when each is computed. The effect of the dollar should be small. Look at this simple example.
Id (the index in dollars) = C * Ie (index in foreign currency)
C= conversion rate. Taking variances and partial derivatives, you get var(Id) = Ie^2*var(C) + C^2*var*(Ie)
For an index value of Ie=1000, exchange rate c=0.5 (roughly dollars per pound), standard deviation(c)=.001 and std dev (Ie) = 10 points, the effect of var(C) is nothing relative to that of var(Ie). Of course you need to do this for all the currencies involved, but you get the point..