I was hoping someone can explain their best technique to approach this type of question – and especially explain the issues I am having with the solution. I have posted both question and solution below.
Here is how far I can get:
1. Calculating Arbitrage-free price: Easy!. I get $1.0183 and as it is lower than the dealer quote, there is an arbitrage opportunity. I even understand that the best scenario is to short the dealer quote and buy euro now.
Then comes my problem:
2. Loan/Risk Free rates: Solution says: Take $1.0231/(1.05)180/365=$0.9988. Why does it use the euro risk-free rate for this USD amount? And then it converts to EURO? I struggle understanding what is actually happening here. Wouldn’t it make more sense to borrow at the lower rate?
Question + Solution
RE: CFA End of Chapter Questions – Derivatives.
12. The euro currently trades at $1.0231. The dollar risk-free 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.
First calculate the fair value or arbitrage-free price of the forward contract:
S0 = $1.0231
T = 180/365
r = 0.04
rf = 0.05
F(0,T) = [1.0231/(1.05)180/365] X [(1.04)180/365) = F(0,T)=$1.0183
The dealer quote for the forward contract is $1.0225; thus, the forward contract is overpriced. To earn a risk-free proﬁt, you should enter into a forward contract to sell euros forward in six months at $1.0225. At the same time, buy euros now.
Take $1.0231/(1.05)180/365=$0.9988. Use it to buy 1/(1.05)180/365= €0.9762 euros.
Enter a forward contract to deliver €1.00 at $1.0225 in six months.
Invest €0.9762 for six months at 5 percent per year and receive €0.9762 × 1.05180/365 =€1.00 at the end of six months.
At expiration, deliver the euro and receive $1.0225. Return over six months is $1.0225/$0.9988 – 1 = 0.0237, or 4.74 percent a year.
This risk-free annual return of 4.74 percent exceeds the US risk-free rate of 4 percent.
I have posted this same message on Prof. Gordon’s Forum and wanted to share his response with you – it really helped me understand the topic better.
By the way, Prof. Gordon (From Examsuccess.ca) runs an excellent CFA Review Program for all levels. I strongly recommend checking it out. He helped me tackle L1 and now is helping with L2.
Answer posted by Prof.Gordon (September 3,2015 14:13:28) – On ExamSuccess.ca/Forum
Here a couple of thoughts to help make it easier:
F = S (1+i$) / (1+i€) <== IRP equation, we know this…make sure you are consistent: since the given quote is $/€
> S (1+i$) / (1+i€) <== this tells us our buy sell: buy the
“cheap” side and sell the “expensive” side, buy the SPOT, sell the
Which currencies do we buy and sell? Rearrange the IRP equation as:
F (1+i€) > S (1+i$) <== borrow the “cheap” side and invest in the “expensive” side, borrow $ and invest €
To address your question 2….
Don’t think of it as a US dollar amount divided by euro interest rate….think of it in terms of the IRP formula:
F / (1+i$) = S / (1+i€)
It is just the “spot exchange rate” divided by the euro interest rate…
The key to all of this is to make sure you line up the quote correctly with the correct “price currency” and “base currency”.
I hope this helps!
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