Preparing for A Summer Vacation(and what it says about arbitrage)

Roberto ChangJanuary 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

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€)

orF€/$ = 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$ = 0.0015, i€ = 0.00105, so the forward rate should be:

F€/$ = 0.75* (1.00105)/(1.0015) = 0.7496