Question

Suppose that interest rate is 9.0 percent per annum in the Canada and 7.0 percent per annum in France, and that the spot exchange rate is $1.50/€ and the forward exchange rate, with one-year maturity, is $1.49/€. Assume that an arbitrager can borrow up to $1,000,000 or €666,666. a) Determine if there is any arbitrage opportunity and the arbitrage profit that can be made if there is. (Show steps of your transactions) b) What would the French interest rate have to be so that there would be no arbitrage opportunity

Answer 1

SOLUTION-

Given that,

Spot rate- EUR 1= USD 1.50

Forward rate- EUR 1 = USD 1.49

Interest rate in Canada = 9%

Interest rate in France = 7%

Calculation of fair forward rate.USD / EUR

EUR 1 = USD 1.50 ( 1+ 0.09 ) / ( 1 + 0.07 )

EUR 1 = USD 1.5280

Since actual forward rate for EUR is less than the fair forward rate . Therefore , EUR is underpriced in forward market.

Hence there is an arbitrage opportunity and to make arbitrage gain the arbitrageur shall buy EUR in forward market.

STEPS FOR ARBITRAGE

STEP 1 - Borrow EUR 666,666 and buy USD.

USD Received = EUR 666,666 * USD 1.50 / EUR

= USD 999,999

STEP 2 - Invest USD 999,999 @ 9% p,a

STEP 3 - After one year, maturity amount received.

USD Received = USD 999,999 ( 1.09)

= USD 1,089,998.91

STEP 4 - Repay EUR Borrowed amount by executing forward contract.

Amount to be repaid = EUR 666,666 (1.07)

= EUR 713,332.62

Buy EUR 713,332.62 by executing forward contract.

USD Outflow = EUR 713,332.62 * USD 1.49 / EUR

= USD 1,062,865.60

Therefore

Arbitrage gain = USD 1,089,998.91 - USD 1,062,865.60

= USD 27,133.31

B)

Calculation of france interest rate so that  no arbitrage opportunity.exists.

There will be no arbitrage opportunity when fair forward rate is equal to the actual forward rate . Therefore when fair forward rate is equal to $ 1.49 / EUR then there will be no arbitrage opportunity.

Ler r be the France interest rate per annum .

1.49 = 1.50 ( 1.09 / 1 + r )

1 + r = 1.50 * 1.09 / 1.49

1 + r = 1.0973

r = 1.0973 - 1

r = 0.0973 or 9.73 %

Therefore when the France interest rate is 9.73% p.a then there will be no arbitrage opportunity.