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I'm currently reading the book "Mastering Attribution" from Andrew Colin.

He first explains that you can separate the FX returns from the security return if the returns are continous compounding:

Assuming continuous compounding, the base currency return $r_{BASE}$ of an asset is given by $$r_{BASE}=r_{LOCAL}+r_{FX}$$ where $r_{LOCAL}$ is its local currency return, and $r_{FX}$ is the foreign exchange return due to changes in the exchange rate between the local and base currencies.

Then he explains that we can separate a portfolio in two subportfolios:

For a clearer view of the effects of the manager’s investment decisions, one should treat this multi-currency portfolio as two subportfolios, one containing securities with returns measured in local currency and one containing FX positions. This allows the value added by local currency investment decisions to be separated from the value added by exchange rate movements.

From what I understand the fact that you can separate the effects is a consequence from the fact that the returns are compounded. If so, why he can add the returns multiplied by the weights if the returns are not arithmetical?

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    $\begingroup$ I believe you can always separate the effect of currency. If returns are continuous you have $r_{BASE}=r_{LOCAL}+r_{FX}$, if they are arithmetic you have $1+r_{BASE}=(1+r_{LOCAL})(1+r_{FX})$ $\endgroup$
    – nbbo2
    Commented Nov 17, 2021 at 21:13
  • $\begingroup$ @noob2 the problem is in that way you can't separe them additively in the arithmetic performance attribution $\endgroup$
    – rlartiga
    Commented Nov 18, 2021 at 12:50
  • $\begingroup$ Yes, that is a big advantage of the continuous compounding method. But what is your question, then. $\endgroup$
    – nbbo2
    Commented Nov 18, 2021 at 14:24

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