Question

Suppose that the annual interest rate is 2.47 percent in the United States and 4.25 percent in Germany, and that the spot exchange rate is $1.60/€ and the forward exchange rate, with one-year maturity, is $1.58/€. Assume that an arbitrager can borrow up to $2,750,000 or €1,718,750. If an astute trader finds an arbitrage, what is the net profit in one year?

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An Italian currency dealer has good credit and can borrow €937,500 for one year. The one-year interest rate in the U.S. is i$ = 2.19% and in the euro zone the one-year interest rate is i€ = 6.089%. The spot exchange rate is $1.25 = €1.00 and the one-year forward exchange rate is $1.20 = €1.00. Show how to realize and calculate the certain euro-denominated profit via covered interest arbitrage:

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You are Microsoft's CFO and have an extra U.S. $1B to invest for six months. You are considering the purchase of U.S. T-bills that yield 1.7975% (that's a six month rate, not an annual rate) and have a maturity of 26 weeks. The spot exchange rate is $1.00 = ¥103.732, and the six month forward rate is $1.00 = ¥111.879. What must the interest rate in Japan (on an investment of comparable risk) be before you are willing to consider investing there for six months?

Answer 1

Part A:

1. Consider the case where you borrow $2,750,000 in the U.S.
2. Given the interest rate of 2.47%, you will be liable to pay back $2,817,925 = $2,750,000*(1+ 2.47%)
3. Post borrowing in the U.S, you can convert the amount to 1,718,750 Euros considering the spot exchange rate of $1.6/Euro
4. Given the interest rate of 4.25% in Germany, you will receive 1,791,796.875 Euros. = 1,718,750 *(1+ 4.25%)
5. Convert this amount back to dollars, to pay back what is owed. So, converting 1,791,796.875 Euros to Dollars using the 1 year forward exchange rate of $1.58/Euros, we get $2,831,039.063.
6. Given that we have to pay back $2,817,925, we can calculate our arbitrage profit as  $2,831,039.063 - $2,817,925 = $13114.0625

Part B:

1. Consider the case where you borrow 937,500 Euros in Italy.
2. Given the interest rate of 6.089%, you will be liable to pay back 994,584.375 Euros = 937,500*(1+ 6.089%)
3. Post borrowing in Italy, you can convert the amount to $1,171,875 considering the spot exchange rate of $1.25/Euro
4. Given the interest rate of 2.19% in U.S, you will receive $1,197,539.063 = 1,171,875 *(1+ 2.19%)
5. Convert this amount back to Euros, to pay back what is owed. So, converting $1,197,539.063 to Euros using the 1 year forward exchange rate of $1.2/Euros, we get 997,949.218 Euros.
6. Given that we have to pay back 994,584.375 Euros, we can calculate our arbitrage profit as  997,949.218 - 994,584.375 = 3364.84 Euros

Part C:

Interest Rate Parity provides us with the interest rate in a non arbitrage condition. IRP formula is given as follows:

(1+i_d) = S/F*(1+i_f)

where i_d is the domestic interest rate, S is the spot exchange rate, F is the forward exchange rate, i_f is the foreign exchange rate.

So, domestic interest rate is 1.7975%, S is 103.732 Yen/$, F is 111.879 Yen/$.

We can calculate the interest rate in Japan as:

(1+i_d)*F/S = (1+i_f)

i_f = ((1+1.7975%)*(111.879/103.732))-1 = 9.7926%

So, the interest rate in Japan must be atleast 9.7926% to consider investing there.

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