Going to the World Cup(and what it says about arbitrage)

Roberto ChangJanuary 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 24000 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.

• http://www.xe.com/currencycharts/?from=USD&to=BRL&view=5Y

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 24000 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 FBRL/$ be the forward exchange rate.

• Then for the plan to succeed,

x * (1 + i$) * FBRL/$ = BR 24000

that is, x = BRL 24000 / [(1 + i$) * FBRL/$ ]

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 EBRL/$ and the interest rate on BRL deposits is iBRL, we need

z* EBRL/$ *(1+ iBRL) = BRL 24000

z* EBRL/$ *(1+ iBRL) = BRL 24000

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

There is no free lunch!

• Summarizing, there are two ways to plan to have 24000 BRL by July:

x = BRL 24000 / [(1 + i$) * FBRL/$ ]

z = BRL 24000/[EBRL/$ *(1+ iBRL) ]• 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 24000 / [(1 + i$) * FBRL/$ ]

= z = BRL 24000/[EBRL/$ *(1+ iBRL) ]

that is(1 + i$) * FBRL/$ = EBRL/$ *(1+ iBRL)

orFBRL/$ = EBRL/$ *(1+ iBRL)/ (1 + i$)

Covered Interest Parity

• The condition FBRL/$ = EBRL/$ *(1+ iBRL)/ (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, EBRL/$ = 2.4, and (approximately) i$ = 0.001, iBRL = 0.05025, so the forward rate should be:

FBRL/$ = 2.4* (1.05025)/(1.001) = 2.52

Concepts

• Exchange Rates: Spot and Forward

• No Arbitrage

• Interest Parity