Question

How to prove the forward exchange rate is F = [S(1+r_d) / (1+ r_f )] under THE PRINCIPLE OF NO-ARBITRAGE? (S is the spot rate and F the forward rate, and r_f and r_d are foreign currency interest rates and domestic currency interest rates respectively) (Or you can show that F < S(1+r_d) / (1+ r_f ) and   F >  S(1+r_d) / (1+ r_f ) violates the principle of no-arbitrage)

Answer 1

Borrow S units of domestic currency for 1 year

Convert to foreign currency you wll get 1 unit of foreign currency

Invest at foreign rate for 1 year

Lets say after 1 year spot rate is F. As forward is the expected spot rate hence current forward rate will be F as well.

After 1 year
You have to return S*(1+rd) domestic currency
You will get 1*(1+rf) foreign currency which you will convert at the prevailing spot rate of F hence you will get 1*(1+rf)*F domestic currency

If the amount to be returned is not the same as amount you gote back, then there will be arbitrage

Hence,
S*(1+rd)=1*(1+rf)*F
=>F=S*(1+rd)/(1+rf)