Question

1. (No-Arbitrage Condition and Interest Parity Condition) Using the concept of no-arbitrage, we can compute a condition that a foreign exchange rate has to satisfy in the short run. Exchange rate is a ratio of the values of two currencies such as dollar and euro. Denote by E the exchange rate of euro in terms of dollar, that is, a dollar value of 1 euro. For example, if E = 1.1 ($/e), then $220 = e ( 220 E ) = e ( 220 1.1 ) = e200, or e100 = $(100×E) = $(100×1.1) = $110 that is, if you want to convert dollar-values to euro-values, you could divide the dollarvalue by the exchange rate, and if you want to convert euro-values to dollar-values, you could multiply the euro-values with the exchange rate. No-arbitrage condition between domestic bond and foreign bond gives a condition called “interest parity condition,” which is an important relationship determining the movement of exchange rates. Let “today” be period t and “next year” be period t + 1. Also assume that the interest rate of U.S. bond is given by i US = 0.1, the interest rate of EU bond is i EU = 0.3, and today and next year’s exchange rates are Et = 1 ($/e) and Et+1 = 0.5 ($/e). Please answer the following questions. (a) Suppose you purchase $100 U.S. bonds today. What is your revenue from the bond (=principal + interest payment) next year? 1 (b) In order to know the “dollar-denominated revenue” from the EU bond, follow the procedure below: i. First, using today’s exchange rate Et = 1 ($/e), compute the euro-value of $100. ii. Next compute the revenue from the EU bond if you invest all the euros you computed in i. iii. Finally, using the next year’s exchange rate Et+1 = 0.5 ($/e), compute the dollar-value of the revenue from the EU bond computed in ii, which is the “dollar-denominated revenue” from investing on the EU bond. iv. Comparing the revenues from the two bonds, which is the better asset to invest on? Do these two bonds satisfy no-arbitrage condition? (c) Now let i US = 0.3 and Et+1 = 1 ($/e) (other parameters are unchanged). Compute the dollar-values of the revenues from U.S. and EU bonds, and show that noarbitrage condition is satisfied. Remark 1. This no-arbitrage condition dollar-revenue from U.S. bond = dollar-revenue from EU bond is called the interest parity condition for dollar and euro

Answer 1

a) Revenue from the bond = $100 +.1*$100 = $110

b) Euro value of $100 = 1*100e = 100e

Revenue from bond (in Euro) = 100e +.3*100e = 130e

Converting Euro revenue into $ = 130e*$.5 = $65

No, these bonds don't satisfy no-arbitrage condition. Comparing (a) and (b), investment in $ denomination is more rewarding by $45 ($110-$65).

c) Revenue from the bond = $100 +.3*$100 = $130

Euro value of $100 = 1*100e = 100e

Revenue from bond (in Euro) = 100e +.3*100e = 130e

Converting Euro revenue into $ = 130e*$1 = $130

In this case, no-arbitrage condition is satisfied.