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

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.

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

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  1. Preparingfor A SummerVacation(and whatitsaysaboutarbitrage) Roberto Chang January 2013 Econ 336

  2. The “problem” • My son and a friend of his are graduatingfromhighschool and savingfor a bigcelebratory “Euro Trip”

  3. Theexchangeratequestion • Theysaytheywillneed, say, about 1000 Euros each, byJuly (sixmonthsfromnow). • Sincethedollar/Euro exchangerate can move a lot, they are wonderingwhatisthebestwayto plan tohavethatamountfortheJulytrip.

  4. Coveringwith a forward contract • A forward contractisanagreementtoexchangecurrencies at a given date in thefuture, at a givenprice (theforward rate) • So, onewaytohave € 2000 in sixmonthsisto set asidesomeamount of dollars (say, x) in aninterestbearingaccount and enter a forward contracttoexchange x*(1 + i$) dollarsfor Euros in July

  5. LetF€/$ be the forward exchangerate. • Thenforthe plan tosucceed, x * (1 + i$) * F€/$ = € 2000 thatis, x = € 2000 / [(1 + i$) * F€/$ ]

  6. Isthereanotherway? • Thereisanalternative: my son couldtakesomeamount of dollarstoday, say y dollars, exchangethemfor Euros today, and savethe Euros in aninterestbearing Euro account • Ifthe (spot) exchangeratetoday (Euros per dollar) isE€/$ and theinterestrateon Euro depositsis i€, weneed z* E€/$ *(1+ i€) = € 2000

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

  8. Thereis no free lunch! • Summarizing, there are twowaysto plan tohavetwothousand Euros byJuly: x = € 2000 / [(1 + i$) * F€/$ ] z = € 2000/[E€/$ *(1+ i€) ] • But x and z must be equal!! • Why? Suppose x < z. Thenbyborrowingthe € 2000, obtaining z dollarstoday, and investing x in dollars, onewouldmake z – x.

  9. Itfollowsthat no arbitragerequires: x = € 2000 / [(1 + i$) * F€/$ ] = z = € 2000/[E€/$ *(1+ i€) ] thatis (1 + i$) * F€/$ = E€/$ *(1+ i€) or F€/$ = E€/$ *(1+ i€)/ (1 + i$)

  10. CoveredInterestParity • Thecondition F€/$ = E€/$ *(1+ i€)/ (1 + i$) isknown as coveredinterestparity. As seen, itisanimplication of no arbitrage. • This can be usedtoinferthe forward exchangerate. Today, E€/$ = 0.75, i$ = 0.0015, i€ = 0.00105, so the forward rateshould be: F€/$ = 0.75* (1.00105)/(1.0015) = 0.7496

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