Question

How much is your arbitrage profit in Euros at expiration, if you know that the current exchange rate is Euro 1.2 / GBP, the 60-day forward rate is Euro 1.16 /GBP, the risk free rate in Europe is 2% and the risk free rate in the UK is 4%? Make your calculation so that the spot transaction is for GBP 500,000.

Answer 1

Covered interest rate parity:

No arbitrage 60-day forward rate, F = 1.2 * [ { e^(2% * (60/360) } / { e^(4% * (60/360) }]

• F = 1.2 * [ { e ^ 0.0033 / e ^ 0.0067 } ]
• F = 1.2 * [ 0.996672 ]
• F = 1.1960

For no arbitrage to hold, 60-day forward rate must be equal to 1gbp = euro 1.1960

However, quoted 60-day forward rate is equal to 1gbp = euro 1.16

There exists arbitrage opportunity and GBP can be bought at lower price from the quoted forward rate at 1.16 euro. So we have to long the 60 day forward contract to buy 1 gbp at 1.16 euro.

At t=0:

• Borrow GBP 500,000 at 4% rate for 60-days
• Convert the borrowed GBP 500,000 to euros at spot rate 1 gbp = 1.2 euro and get 500,000 * 1.2 = 600,000 euros
• Invest 600,000 euros at 2% for 60-days.
• Enter 60-day forward contract to buy 1 gbp at 1.16 euros

After 60 days:

• Receive investment proceeds in Euros of amount equal to 600,000 * e ^ (2% * (60/360)) = 600000 * 1.003339 = 602,003.30
• Oblige forward contract and buy 1 gbp for 1.16 euros and receive total GBP equal to 602,003.30/1.16 = 518,968.40 GBP
• Repay amount borrowed in GBP = 500,000 * e ^ (4% * (60/360)) = 500,000 * e ^ (0.00667) = 500,000 * 1.006689 = 503,344.50
• Amount left = 518,968.40 GBP - 503,344.50 GBP = GBP 15,623.92