Going to the World Cup (and what it says about arbitrage) Roberto Chang January 2014 Econ 336

The “problem” A number of people I know are thinking about going to Brazil for the World Cup It is very expensive, so we need to make efficient financing decisions

The exchange rate question They say they will need, say, about Brazilian reais (BRL) each, by July (six months from now). Friday’s spot exchange rate: 2.40 BRL per US$ So, at current rates, the amount involved is about US $ 10,000 But the BRL/US$ exchange rate can move a lot, we 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 BRL in six months is to set aside today some amount of dollars (say, x) in an interest bearing account and enter a forward contract to exchange x*(1 + i $ ) dollars for reais in July

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

Is there a cheaper way? There is an alternative: one could take some amount of dollars today, say z dollars, exchange them for reais today, and save the reais in an interest bearing BRL account If the (spot) exchange rate today (reais per dollar) is E BRL/$ and the interest rate on BRL deposits is i BRL, we need z* E BRL/$ *(1+ i BRL ) = BRL 24000

Or, equivalently, z = BRL 24000/[E BRL/$ *(1+ i BRL ) ]

There is no free lunch! Summarizing, there are two ways to plan to have BRL by July: x = BRL / [(1 + i $ ) * F BRL/$ ] z = BRL 24000/[E BRL/$ *(1+ i BRL ) ] But x and z must be equal!! Why? Suppose x < z. Then by borrowing the BRL 24000, obtaining z dollars today, and investing x in dollars, one would make z – x instantly, at no cost, and without risk.

Implications of No Arbitrage It follows that no arbitrage requires: x = BRL / [(1 + i $ ) * F BRL/$ ] = z = BRL 24000/[E BRL/$ *(1+ i BRL ) ] that is (1 + i $ ) * F BRL/$ = E BRL/$ *(1+ i BRL ) or F BRL/$ = E BRL/$ *(1+ i BRL )/ (1 + i $ )

Covered Interest Parity The condition F BRL/$ = E BRL/$ *(1+ i BRL )/ (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 BRL/$ = 2.4, and (approximately) i $ = 0.001, i BRL = , so the forward rate should be: F BRL/$ = 2.4* ( )/(1.001) = 2.52

Concepts Exchange Rates: Spot and Forward No Arbitrage Interest Parity