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Homework answers / question archive / Suppose that six-month interest rates (annualized) in Japan and the United States are 7 percent and 9 percent, respectively
Suppose that six-month interest rates (annualized) in Japan and the United States are 7 percent and 9 percent, respectively. If the spot rate is ¥142:$1 and the ninety-day forward rate is ¥139:$1 and the 180 day forward rate is: ¥152:$1.
What arbitrage opportunity do these figures present?
And assuming no transaction costs, what would be the arbitrage profit per dollar or dollar-equivalent borrowed?
For simplicity, let's assume that there is continuous compounding. Given these values the following arbitrage opportunity exists:
- Using the 180-day forward contract, agree to sell $1 in exchange for ¥152 180 days from now
- Borrow ¥142 today and use it to buy $1 (that's the spot rate)
In 6 months, you'll hand over the $1 and receive the ¥152. At that time, since you initially borrowed ¥142, you will owe 142*exp(0.07*0.5) = ¥147.05. Also, since you can deposit the $1 you bought, you would have 1*exp(0.09*0.5) = $1.046 in 6 months.
Now, after you give the $1, you'll receive ¥152. After using ¥147.05 to repay your debt, you will be left with a profit of 152 - 147.05 = ¥4.95. You will also have the extra $0.046, which comes from depositing the $1 for 6 months. The sum of these two is the risk-free profit.
At the spot rate, we have that ¥4.95 = $0.0348. Therefore, the arbitrage profit per dollar-equivalent borrowed would be 0.0348 + 0.046 = $0.0808.
