The following information is available: Spot rate for Euro: $1.4/€; 511-days fut
ID: 2779838 • Letter: T
Question
The following information is available: Spot rate for Euro: $1.4/€; 511-days futures rate for Euro: $1.50/€ (assume 365-day year); U.S. risk-free rate: 3%; Euro risk-free rate: 5%.
a. A U.S. investor who can borrow in dollars (look at it from U.S. perspective) is looking at the above quotes. Find for this U.S. investor the correct futures price for the 511-days futures on $/€ exchange rate using continuous compounding, and then list the specific transactions in the spot, futures, and credit markets needed to avail an arbitrage opportunity if possible, and calculate the arbitrage profit per Euro. For arbitrage transactions, assume that U.S. investor can borrow $1,000 at 3%.
b. Now think of a European investor who can borrow in Euros. He treats the U.S. risk-free rate as the foreign interest rate (q variable) and his spot and futures rates are €0.7143/$ and €0.6667/$, respectively (i.e., inverse of the above $/€ exchange rates). Find for this European investor the correct futures price for the 511-days futures €/$ exchange rate using continuous compounding and show possible arbitrage transactions (Assume that European investor can borrow €714.30 at 5%.).
Explanation / Answer
a:
S = 1.4 USD/EUR, F = 1.5 USD/EUR
Expected future rate
F = S*exp[(r – q)*t]
When exchange rates are in USD/EUR, r and q are in USD and EUR.
F = 1.4*exp((0.03 – 0.05)*511/365) = 1.36
As expected future rate is not same as given future rate, there will be an arbitrage opportunity.
Borrow 1 USD at USD rate and invest at EUR rate
It needed to close loan in USD after 511 days 1*exp(0.03*511/365) = 1.04
Investment becomes after converted in to USD =
1*(1/1.4)*exp(0.05*511/365)*1.5 = 1.15 USD
Profit = 1.15 – 1.04 = 0.11 USD per 1 USD
For 1000 USD, profit = 110 USD after rounding.
b:
S = €0.7143/$, F = €0.6667/$
Expected future rate = 0.7143*exp((0.05 – 0.03)*511/365) = 0.7346 EUR/USD
Now, borrow €714.30 at EURO rate and invest in USD
Loan accumulated value = 714.30*exp(0.05*511/365) = 766.09 EUR
Convert borrowed into USD, invest in USD, and again convert it into EUR
Final value = 714.30*(1/0.7143)*exp(0.03*511/365)*0.6667 = 695.30 EUR
If you borrowed in EUR and invest in USD, you need more money to close the loan than the investment grows. There would be no profit. To get arbitrage profit, you need to borrow in USD for given arrangement.
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