Assume you observe the following information: Bid Ask Spot ($/Euros): 1.4000 1.4

Question

Assume you observe the following information:

Bid   Ask

Spot ($/Euros):    1.4000 1.4300

360-day Forward ($/Euros): 1.4400 1.4900

U.S. Interest Rate:     4.10% 4.20%

Please note that the bid U.S. interest rate is the deposit (or lending rate) a bank would pay a customer for depositing funds in the bank,

while the ask interest rate in the U.S. is the borrowing rate a customer can borrow at from a bank.

How high does the lending rate in the euro zone have to be before an arbitrageur would NOT consider borrowing dollars, trading for euro at the spot, investing in the euro zone and hedging with a short position in the forward contract?

4.123%

3.176%

3.665%

3.478%

4.123%

3.176%

3.665%

3.478%

Explanation / Answer

Let us consider an arbitrageur borrows $ 10000

Intrest charged = 4.20 % (given in the question)

Amount to be paid to bank after one year = 10000 * 1.0420 = $ 10420

Arbitrageur will buy Euro from spot using $ 10000 at spot rate of 1.43 $/Euro

No of Euros he will get = 10000 / 1.43 = 6993.00 Euros

Now he will invest this Euros to euro zone for some percentage return. Since we are supposed to find the lending rate so that there is no arbitrageur opportunity. For no arbitrageur opportunity amount he needs to pay to American bank should exactly match with the amount he get after converting intrest earned Euros back to $ at forward rate.

So let as assume lending rate in Euro zone is X %

Euros he will get after depositing 6993 Euros at X % = 6993 * ( 1+ X/100)

To convert this Euros to $ he needs to buy $ at 1.44 $/ Euro ( Bid - forward rate given in question )

So, total $ he will have after converting =   6993 * ( 1+ X/100) * 1.44

As explained earlier to avoid arbitrageur expected $ should match with what he owe to American Bank, which implies

10420 = 6993 * ( 1+ X/100) * 1.44

7236.11 =  6993 * ( 1+ X/100)

1.03476 = ( 1+ X/100)

X = 3.476 %

So correct answer is D

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