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How to correctly express basket of currencies in and index, such that P&L would align?

Assume our index is 20% EURUSD and 80% GBPUSD and rates are 1.10 and 1.31 for T1 and 1.05 and 1.35 for T2. On a USD 100,000 total real P&L is 20,000*(1.05/1.10-1) + 80,000*(1.35/1.31-1) = $1,533.66

Now for the index, whatever I try, I can't get $1,533.66...

TRY1 weights

$100,000 * ( (1.05*0.2 + 1.35*0.8)/(1.10*0.2 + 1.31*0.8)-1 ) = $1,735.02

TRY2 permanent weights

$100,000 * ( (1.05^0.2 * 1.35^0.8)/(1.10^0.2 * 1.31^0.8)-1 ) = $1,486.74

It's not miles off, but there must be a way to calculate accurate index, returns of which align to first principles PL calc?

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  • $\begingroup$ Take a look at the numerical example in quant.stackexchange.com/questions/69994/… $\endgroup$
    – Dimitri Vulis
    Mar 21 at 23:30
  • $\begingroup$ Thanks! But question is more about constructing the index rather than calculating P&L. Hw would you construct an 0.2 * EURUSD and 0.8 * GBPUSD Index? $\endgroup$
    – John
    Mar 22 at 6:53
  • $\begingroup$ The "shortcuts" TRY1 and TRY2 do not work. To calculate the index (starting at say 100), you calculate the return as you did in the 2d paragraph as the weighted avg of the currency returns, then you update the index as index(t)=index(t-1)*(1+ret(t)). This document for Bbg Currency Indexes may be of interest assets.bbhub.io/professional/sites/27/… $\endgroup$
    – nbbo2
    Mar 22 at 21:14

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