 # Risk-free arbitrage with currencies

* The Japanese yen currently trades at \$0.00812/Y. * The U.S. risk free rate is 4.5%. * The Japanese risk free rate is 2.0%. * Three month forward contracts on the yen are quoted at \$0.00813/Y. Describe how you can earn a risk-free profit by engaging in a forward contract. The answer says that you get a return of 3.88%, better than the 2% you would get on the Japanese yen. This is problem 15 on page 54 of Derivatives. I have a question, but first let me explain how this problem is approached. 1. Check to see what the fair value of the forward contract is. It will be \$0.00817/yen. 2. So, one yen should cost you \$0.00817. However, the forward market is offering it for \$0.00813. It’s cheaper, so you should buy the forward contract. 3. Buying the forward contract means you deliver \$0.00813 and you get one yen. 4. The solution suggests that you *take* some yen and buy dollars with it. Deposit the dollars at the U.S. rate, and then after 90 days, use the proceeds to execute the contracts and receive the yen. I have a problem with *taking* the yen you use to purchase the dollar. Assuming you start with zero money, you need to borrow those yens, which will cost you interest in 90 days. That is not described by the problem. If you do it this way, you will lose money by engaging in this transcation. So I thought may be this is how the currencies market works: You short the yen and you long the dollar. But even here, I think you still have to pay the interest on the short side. Any help?

First rule of arbitrage - always, always always, use someone else’s money. So borrow and pay interest, invest in something that is risklesss, and pay back loan and you still have a profit. Here is how I would approach this: First, since no bid-ask. Go one way around and see where you end up. If you lose try the other way. Borrow yen enter forward rate in future. To make the math easier borrow 1,000 Yen. You then owe 1.02^(90/360)*1000 = 1004.9629 Yen in 90 days. Convert to US @ spot 0.00812 =8.12 earn US interest rate = 8.12*1.045^(90/360) = 8.2098 Now convert to Yen using your forward rate agreement = 8.2098 / .00813= 1009.8155 Pay off loan plus interest and whats left is profit = 1009.8155 -1004.9629 =4.8526 profit If you get a loss then go the opposite direction. The key concept is to realize that you are calculating what the interest rate parity says things should be in the future. If you didn’t have the forward, it would not be riskless arbitrage but since you can enter the forward you are locking-in your conversion back so you can make a riskless arbitrage profit. Make sense?

> Pay off loan plus interest and whats left is profit = 1009.8155 -1004.9629 =4.8526 profit That’s how I did it (which similar to how the book did it), but it is not profitable if you look at the last step. You earned 4.8562 yens, out of 1000 yens, that’s a profit of 0.485% in 90 days, or about 1.94% annually, which is less than 2% the Japanese rate!

The profit is AFTER paying the interest cost. You are double counting if you compare your profit to your loan. Step 1 we are paying 1.02^(90/360) = loan +interest. So the profit above is net the interest cost. The other point. Remember, we didn’t put up 1000 yen (see first rule of arbs in prev post). So we earned 4.8562 on 0 equity capital. That’s the fundamental of arbitrage. You have zero equity to start and end up with profit. We borrowed, so in essence our return is infinite or limited by our ability to borrow.

Re-read your post. I understand where you are coming from but you don’t need to bring in holding period return. It doesn’t work for arbitrage. You start with 0 You borrow 1000 You earn up to 1009.8155 You pay back 1004.9629 Profit 4.8526 So if you tried to calculate a HPR its Ending - Begining / begining (4.8526-0)/0 It approximates infinity.

Nope that doesn’t make sense. Ignoring the zero equity thing for now, the way the book did it was to assume that you pay no interest on the borrowed yen. If you add back the interest charge, you would get the exact answer in the book. But I humbly think the book is oversimplifying this concept by making a wrong assumption.

looking at the book they handle derivative different than I typically do. They did NOT do this step: 1.02^(90/360)*1000 You and I both did, so our answer anything great than 0 is good. So their calcs do not calculate interest. Seems like an odd way to do it. Their method just seems confusing. It appears they are calculating the return. So in my example its appears its 1009 (the amount of yen we end up with)/1000 (amount of yen we started with)-1 which is their approx .00969 which annualized is 3.8 approximately. So they get the same answer, thats just a confusing way to do it. I see why this is confusing in the book. Let me know if my explanation is bad.

In your method and mine, there is no arbitrage. You will lose money in this transaction. The book’s method for this particular problem, I don’t know about the other problems, is wrong, it seems. Can anyone else chime in?

I agree with stringreye, I mean the book does it in a confusing way, as the CFAI do everything that way… If you ignore the return side of things, and the last part of their answer, basically you start out with 0, and you end up with 4.85, that 4.85 is risk free and therefore arbitrage. But the book didnt take into account the interest you have to pay back on the loan, if you do then you end up with an annualised return of roughly 2% I think. What they do is compare your return before you pay back the yen interest, so they are comparing 3.88% with 2%, and they are saying because it exceeds the interest cost you have made a profit. The after interest return is then 1.88%, which is still a positive return

stingreye Wrote: ------------------------------------------------------- > You start with 0 > You borrow 1000 > You earn up to 1009.8155 > You pay back 1004.9629 > Profit 4.8526 > > So if you tried to calculate a HPR its > Ending - Begining / begining > > (4.8526-0)/0 > It approximates infinity. It’s not correct to view it like that. Here an investment of 1000 yen turned into 1004.9629 yen, a return of 0.496%. Whether those 1000 yen were yours or your sister’s is immaterial. No such thing as infinite return.

pedpenny Wrote: ------------------------------------------------------- > a profit. The after interest return is then 1.88%, > which is still a positive return I agree that if you *lock in* any rate *above* the avaiable rate, that would be a risk-free profit. But i’m not sure the above situation is risk-free. Here is another more basic problem as an example. You borrow \$100k from your broker at 9% interest. You buy IBM shares. A year later you sell IBM for \$110k. You pay back the broker \$109k, and you pocket \$1000. Was that risk-free? Of course not. Did you earn a higher rate than a bank CD (of say 3%)? Well, what was your return? It’s not \$1000/0=infinity! Your return is \$1000/\$100k= 1%. An investment of \$100k turned into \$101k.

Well i agree that your above trade is not risk free, but the difference between buying a stock and buying a forward contract is the price you get in the future is fixed, unless the counterparty defaults and then your screwed so its not entirely risk free, but lets face it would could spend the day arguing that alot of the stuff the cfa teachs is you is not entirely applicable to the real world, thats because alot of it is just financial theory, and may not hold in practice, but its the CFAI’s exam and we just have to accept that for the exam