Question 23.3: Suppose that in December 2016, the spot exchange rate for th...
Suppose that in December 2016, the spot exchange rate for the Japanese yen is ¥116/$. At the same time, the one-year interest rate in the United States is 4.85% and the one-year interest rate in Japan is 0.10%. Based on these rates, what forward exchange rate is consistent with no arbitrage?
PLAN
We can compute the forward exchange rate using Eq. 23.1. Because the exchange rate is in terms of ¥/$, we need to make sure we are dividing 1 plus the yen rate by 1 plus the dollar rate:
\underbrace{\text{Forward Rate} }_{\frac{€ \text{ in one year}}{\$ \text{ in one year}} } = \underbrace{\text{Spot Rate}}_{\frac{€ \text{ today}}{\$ \text{ today}} } \times \underbrace{ \frac{(1+r_€)}{(1+r_\$)}}_{\frac{€ \text{ in one year /€ today}}{\$ \text{ in one year /\$ today}} } (23.1)
\text{Forward Rate}_{¥/\$}= \text{Spot Rate}_{¥/\$} \times \frac{(1+r_¥)}{(1+r_\$)}(A useful rule to remember is that the ratio of interest rates must match the units of the exchange rate. Because the exchange rate is ¥/$, we multiply by the yen interest rate and divide by the dollar interest rate. We could also solve the problem by converting all the rates to $/¥.)
EXECUTE
\text{Forward Rate}_{¥/\$}= \text{Spot Rate}_{¥/\$} \times \frac{(1+r_¥)}{(1+r_\$)} = ¥116/\$ \times \frac{1.0010}{1.0485} = ¥110.7/$ in one year
EVALUATE
The forward exchange rate is lower than the spot exchange rate, offsetting the higher interest rate on dollar investments. If the forward exchange rate were anything other than ¥110.7/$, such as ¥111/$, arbitrage profits would be available. We could have borrowed ¥1 billion at 0.10% interest, exchanged it into $8,620,690 (¥1 billion ÷ ¥116/$ ) and deposited the dollars, earning 4.85% interest. In one year, we would have $9,038,793 ($8,620,690 × (1.0485)) and owe ¥1.0010 billion (¥1 billion × (1.0010) ). If we had locked in a forward exchange rate of ¥111/$, we would need $9,018,018 (¥1.0010 billion ¥111/$) to pay off our loan, leaving us with a no-risk profit of $20,775! We (and everyone else) would do this until the forward rate came into line with the no-arbitrage rate of ¥110.7/$.