Calculating return in domestic currency (Topic: Case for international diversification)

[anks612]

According to the book, the formula for return in domestic currency is:
R($) = R(LC) + S + [R(LC)]*S
where
R ($) = return on the foreign asset in U.S. dollar terms
R(LC) = return on the foreign asset in local currency terms
S = percentage change in the foreign currency
Can anyone explain why the third term (ie [R(LC)]*S) needs to be added? Why isn’t the formula simply R($) = R(LC) + S ?

 

[QNA2012]

base/domestic return = local/foreign asset return + currency return
this equation is only accounting for the principal investment and ignoring the change in the principal investment
the extra term is to account for the growth or decline in principal investment so that:
Rd = Rf + s (1 + Rf) = Rf + s + sRf
the second term s(1 +Rf) means you are translating the foreign currency investment at the end of your investment period (or valuation date) back to base/reporting currency

 

[BMiller12]

I wondered the same thing initially. In my mind, it is similar to the difference between a wealth tax (which taxes principal and gain) vs a capital gains tax which just taxes the gain. Here, accounting for the change in currency is more akin to the wealth tax.
S is really like accounting for the change in the principal and the third term is accounting for the currency change to the return (gain). The sum of the two ensures you are capturing the currency change in principal and the return.
Someone let me know or chime in if i’m mistaken or this analogy is poorly founded.

 

[anks612]

@QNA2012 and BMiller12
Thanks guys…that really cleared it up.

 

[1recho]

Pretty much all returns composed of two or more returns are like that. e.g.
nominal return vs real return and inflation:
1 + n = (1 + r) (1 + i) = 1 + r + i + ri
n = r + i(1+r)
In this case
1 + R($) = (1 + R(LC)) (1 + S)
If you had bought the local asset using 1 local currency then it would be worth 1 + R(LC) right now
If you had bought $1 worth of local currency then it would be worth $(1 + S) right now.
You went ahead and bought $1 worth of local asset. Without any price appreciation that would be worth $(1 + S) just because of currency fluctuation. But the asset itself increased (hopefully) in worth by its local return so it is 1 + R(LC) in local units and hence, (1 + R(LC)) * (1 + S) in $.