Question

The spot exchange rate today is 1.32 US Dollars for every Euro. Suppose the 6-month continuously compounded interest rates are 2% in the US and 3% in Europe.

(a) What should the price of a currency futures contract deliverable in 6 months be?
(b) Suppose that the futures price quoted in the market is 1.30. What would you do to profit from the situation? Is it an arbitrage?

Hint: Long a futures contract (for the quoted futures price), lend out some amount of US Dollars and borrow the equivalent amount in Euros. Choose this amount in such a way that you have E100 at the futures contract delivery. What is the difference in US Dollar terms of the proceeds of the two loans and the futures contract?

Answer 1

a) Spot exchange rate = 1.32 USD/EURO

Interest rate = 2% in US and 3% in EUROPE

As per the Interest rate parity, the forward (future price) and spot price are related by the equation

Future exchange rate/Spot exchange rate =e^[(domestic interest rate - foreign interest rate) *  n ]

where n = no.of years = 0.5 years (6 months)

So, Futures exchange rate after 6 months = 1.32 * e^[(0.02-0.03)*0.5] = 1.32*e^(-0.005) =1.3134

So, price of a currency futures contract deliverable in 6 months should be 1.3134 USD/Euro

b) If the futures price quoted is 1.30 , then the profit can be obtained by purchasing the futures contract (purchasing Euro at 1.30 after 6 months by selling USD)

If Euro 100 is the target , then one should have Euro 100 * 1.30 USD/Euro = USD 130 at the end of 6 months

So, today one must lend USD @2% for an amount of 130*e^(-0.02*0.5) = $128.7065

This amount can be gathered by borrowing Euro equivalent to 128.7065/1.32 =Euro 97.5049 today @3%

and repaying 97.5049*e^(0.03*0.5)= Euro 98.9785 after 6months

So, the steps are as follows

1. Borrow Euro 97.5049 today and convert it to USD128.7065 today as per calculations above

2. Lend USD 128.7065 @ 2% for 6 months today

3. Purchase the currency futures contract worth Euro 100 today (no money needed)

4.After 6 months. get USD 130 from the amount lent and use the futures to get Euro 100

5.Repay borrowed Euro loan along with interest for a total of Euro 98.9785 and pocket the remaining Euro 1.0215 as arbitrage profit (riskless profit).

Difference in USD terms for the amount lent = 130 - 128.7065 =USD 1.2935

Difference in USD terms for the amount borrowed = Euro 98.9785 * 1.32 USD/Euro - Euro 97.5049 *1.30 USD/Euro

= - USD 0.03442

hence, the total benefit = USD benefit in amount lent - USD loss in amount borrowed   

= USD1.2935 - (-USD 0.03442)

= USD 1.3279

=Euro 1.3279 /1.3

= Euro 1.0215 which is the same as above