Using forward rate to get risk free arbitrage

CFA Level I
#1

Hi everyone,

I am having troubles with this question. Any suggestions how to solve the problem are very welcome! Thank you!

The spot rate USD/JPY is 80,3. 9M forward rate is 81,5. The USD interest rate is 2%, Japan interest rate is 1,5%

a) Use the forward rate to get the risk free arbitrage profit if US investor has 1 Mill of USD.

#2

The fact that the investor has USD1,000,000 isn’t relevant. For arbitrage, the investor must borrow money; otherwise, the investor has money at risk.

Presuming that the spot rate of 80.3 means that 80.3¥ = $1.00 (which means that your quote is opposite of CFA Institute’s convention), and that the interest rates are effective, annual rates, the arbitrage-free 9-month forward rate is 80.3 × (1.015/1.02)^(9/12) = 80.0046. As this is less than the quoted 9-month forward rate, the investor wants to begin the arbitrage transaction by borrowing yen (because the forward rate pays too many yen per dollar; you always want to get paid too much). Here’s the sequence of transactions:

  1. Borrow, say, JPY100,000,000 for 9 months at 1.5%.
  2. Convert JPY100,000,000 to USD1,245,330 at the spot rate of 80.3 JPY per USD.
  3. Enter into a forward contract to convert USD1,263,964 to JPY in 9 months at 81.5 JPY per USD.
  4. Invest USD1,245,330 for 9 months at 2.0%.
  5. Wait 9 months. Your investment will grow to USD1,263,964 (= USD1,245,330 × 1.02^(9/12)).
  6. Convert USD1,263,964 to JPY103,013,037 at the forward rate of 81.5 JPY per USD.
  7. Pay off the JPY loan for JPY101,122,904 (= JPY100,000,000 × 1.015^(9/12)).
  8. Reap a profit of JPY1,890,133.
#3

Thank you very much for your help! :slight_smile:

#4

My pleasure.