Question

Assume 1 year interest rate on risk free securities in the united states is 10 %, and in UK is 6% and the current dollar/£ rate is 1.5 dollar.

A. What do you expect the forward exchange rate in dollar or pound for 12 months to be today?

B. Would the forward exchange rate be equal to the future spot exchange rate every day in the year? Why?

Answer 1

We will use interest rate parity equation to arrive at the forward exchange rate for 12 months hence, by using the formula: Forward Rate = Spot Rate * (1+ r_domestic) / (1+ r_other) ; where r is the interest rate in domestic and other country. Now we are give Spot Rate (S) is 1GBP = 1.50 USD or 1USD = 0.6667 GBP, r_USD = 10% and r_GBP = 6%. Hence Forward Rate (F) = 1.50 * (1+10%) / (1+6%) = 1.5566

Theoretically, for a no arbitrage condition to be satisfied, the forward exchange rate should be equal to the future spot rate every day of the year because in case of a mismatch, traders will be able to profit from possible arbitrage. However even if there is an arbitrage, the subsequent transaction flow to profit from the arbitrage will change the demand supply dynamics such that the future spot prices are in line with the forward rate.

We can see with an example - let us calculate the forward price after 3 months using the interest rate parity:

F_{3 month} = 1.50 * (1+10%/4) / (1+6%/4) = 1.5148 (rounded to 4 decimals). Now if the future spot price at this time is 1.52, then a trader can do the following:

• GBP is higher than it should be. Hence we borrow GBP 100 at 6% for 9 months. We will have to pay GBP 104.5 after 9 months
• Convert this GBP in to USD and receive USD (100 * 1.52) = 152 which will be placed in USD deposit at 10% for 9 months. After 9 months we will receive USD 163.4
• The forward exchange rate after at end of 12 months (9 months from time of above trade) as calculated above should be 1.5566 which will give GBP (163.4/1.5566) = GBP 104.97
• After paying off the GBP borrowing & interest GBP 104.5, the trader will be left with pure arbitrage profit of (104.97-104.50) GBP 0.47. This arbitrage profit will persist till the forward exchange rate as per interest rate parity and future spot rate are not same .
• Due to this arbitrage the traders will start selling GBP and convert into USD, which will lead the spot rate to equilibrium price at the end of 3 mnths of 1.5148 which is no arbitrage level.

Hence, the forward exchange rate should be equal to the future spot rates every day of the year.