I don’t want to go too far, but it seems that: (Currency Appreciation) is not always equal to -(Currency Depreciation) For example, S0=100 yen/usd, S1=105 yen/usd. USD appreciates by (105-100)/100=5%. YEN depreciates by (1/105-1/100)/(1/100)=-4.76%. Where am I wrong? They should be the same.
because absolute changes do no equal relative changes. This is just a mathematical feature, nothing to do with depreciation of currencies. Just like in the example you gave. Other example: if you go from 100 to 200, then the absolute difference is 100 and the relative difference is 100%. If you go from 100 to 0 then the absolute difference is -100, but the relative difference is - infinity.
you are failing to account for the current exchange rate. when a and b (the two currencies) are originally linked by a factor, e.g. a = x * b when a increases by 5% -> b would only need to increase by 5%/x so the two would not move by the same amount. a = 100 * b so 1.05 a = 100/1.05 * b or 95.23 b – -4.76%