I am having trouble understaning the detailed abitrage transactions in EOC Q12.
Here is the question:
The Euro currently trades at $1.0231. The dollar risk freee rate is 4 percent, and the euro risk free rate is 5 percent. Six month forward contracts are quoted at a rate of $1.0225. Indicate how you might earn a risk free profit by engaging in a forward contract. Clearly outline the steps you undertake to earn this risk free profit.
I understand to get the calculation. The forward rate I calculated is $1.0179, but understand the steps to earn the risk free profit is hard…
Here is the solution from the book:
Take $1.0231 / (1.05)^(180/365) = $0.9988. Use it to buy 1/ (1.05)^(180/365) = 0.9762 euros.
why take $1.0231/(1.05)^(180/365) ? Isnt this a current price already? Also, why 1/(1.05)^(180/365) = euros?
Enter a forward contract to deliver 1 euro at $1.0225 in six months.
Invest euro 0.9762 for six months at 5 percent per year and receive euro 0.9762 * 1.05^(180/365) = euro 1 at the end of six months.
At expiration, deliver the euro and receive $1.0225. Return over six months is 1.0225/0.9988 - 1 = 0.0237, or 4.74% a yr.
They just want to end up with exactly one euro after 180 days. So they start with 0.9762 euro today, so they start with 0.9988 dollars today. There’s nothing special about ending up with one euro in 180 days; they’re making it unnecessarily complicated.
I would start by computing what the forward exchange rate should be:
$1.0231/€ × (1.04/1.05)^(180/365) = $1.018283/€
(I’m not sure how you got $1.0179/€.) Thus, forward euro (at $1.0225/€) are overpriced, so we want to buy (fairly priced) euro today and sell (overpriced) euro in 180 days.
Here are the steps:
Borrow $1 for 180 days at 4%.
Convert $1 to $1 / [$1.0231/€] = €0.977422.
Invest €0.977422 for 180 days at 5%. This will grow to €0.977422 × 1.05^(180/365) = €1.001224 in 180 days.
Enter into a forward contract to deliver €1.001224 in 180 days at the forward rate of $1.0225/€.
Wait (patiently) for 180 days.
Convert €1.001224 to €1.001224 × $1.0225/€ = $1.023752.
Pay off the loan (with interest) for $1 × 1.04^(180/365) = $1.01953.
Revel in your profit of $0.004222 (= $1.023752 – $1.01953): 0.4222% for 180 days, or (1.004222)^(365/180) – 1 = 0.8580% annually.
The book’s calculation doesn’t borrow the dollars initially, so their 4.74% per year isn’t the arbitrage profit; the profit is 4.74% – 4% = 0.74% (over the risk-free rate). Because they’re not borrowing the dollars, it’s not really an arbitrage transaction: they have risk.
From my experience, if you don’t practise these problems, you’ll be in big trouble. Do as many practice problems as you can.
I think derivatives will be a heavily weighted topic, even though the range is 5-15. Plus there is overlap in currency forwards from econ also, so it is a very good idea to know this material inside out.