Preparing for A Summer Vacation (and what it says about arbitrage) Roberto Chang January 2013 Econ 336

The “problem” My son and a friend of his are graduating from high school and saving for a big celebratory “Euro Trip”

The exchange rate question They say they will need, say, about 1000 Euros each, by July (six months from now). Since the dollar/Euro exchange rate can move a lot, they are wondering what is the best way to plan to have that amount for the July trip.

Covering with a forward contract A forward contract is an agreement to exchange currencies at a given date in the future, at a given price (the forward rate) So, one way to have € 2000 in six months is to set aside some amount of dollars (say, x) in an interest bearing account and enter a forward contract to exchange x*(1 + i $ ) dollars for Euros in July

Let F €/$ be the forward exchange rate. Then for the plan to succeed, x * (1 + i $ ) * F €/$ = € 2000 that is, x = € 2000 / [(1 + i $ ) * F €/$ ]

Is there another way? There is an alternative: my son could take some amount of dollars today, say y dollars, exchange them for Euros today, and save the Euros in an interest bearing Euro account If the (spot) exchange rate today (Euros per dollar) is E €/$ and the interest rate on Euro deposits is i €, we need z* E €/$ *(1+ i € ) = € 2000

Or, equivalently, z = € 2000/[E €/$ *(1+ i € ) ]

There is no free lunch! Summarizing, there are two ways to plan to have two thousand Euros by July: x = € 2000 / [(1 + i $ ) * F €/$ ] z = € 2000/[E €/$ *(1+ i € ) ] But x and z must be equal!! Why? Suppose x < z. Then by borrowing the € 2000, obtaining z dollars today, and investing x in dollars, one would make z – x.

It follows that no arbitrage requires: x = € 2000 / [(1 + i $ ) * F €/$ ] = z = € 2000/[E €/$ *(1+ i € ) ] that is (1 + i $ ) * F €/$ = E €/$ *(1+ i € ) or F €/$ = E €/$ *(1+ i € )/ (1 + i $ )

Covered Interest Parity The condition F €/$ = E €/$ *(1+ i € )/ (1 + i $ ) is known as covered interest parity. As seen, it is an implication of no arbitrage. This can be used to infer the forward exchange rate. Today, E €/$ = 0.75, i $ = , i € = , so the forward rate should be: F €/$ = 0.75* ( )/(1.0015) =