Currency forwards and interest rate parity
1. The idea
A dollar investor can hold dollars for a year and earn the dollar interest rate \(r\). Or the investor can change the dollars into euros today, earn the euro interest rate \(r_f\), and change them back in a year at the one-year forward exchange rate, agreed today. Both plans are free of risk, so they must give the same number of dollars, or one would be an arbitrage. That fixes the forward exchange rate, the forward price of one unit of the foreign currency:
\[ F_0 = S_0\, e^{(r - r_f)T}, \]
where \(F_0\) is the forward exchange rate and \(S_0\) the spot exchange rate, the spot price of one unit of the foreign currency, both in dollars, \(r\) the dollar interest rate, \(r_f\) the foreign one, both with continuous compounding, and \(T\) the time to delivery in years. This is interest rate parity.
An example. One euro costs $1.1000, the dollar interest rate is 4.5% and the euro interest rate 2.5%. The one-year forward exchange rate is \(1.1000 \times e^{(0.045 - 0.025) \times 1} = \$1.1222\). The euro’s forward exchange rate is above its spot exchange rate because the euro interest rate is lower than the dollar interest rate: whoever holds euros for the year earns two percentage points less interest, and the higher forward exchange rate makes up for exactly that.
So the forward exchange rate of the currency with the lower interest rate is above its spot exchange rate, and that of the currency with the higher interest rate is below it. The forward exchange rate is not a forecast of the spot exchange rate at delivery; it is only the rate that leaves no arbitrage. A foreign currency earns its interest rate \(r_f\) the way a share earns dividends; the cost of carry treats both as income that lowers the forward price. Turned around, the forward and spot exchange rates give the difference between the two interest rates, \(r - r_f = \ln(F_0/S_0)/T\), where \(\ln\) is the natural logarithm.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The forward exchange rate on a chart
First, which way the forward exchange rate moves.
Now move the delivery date. The further away it is, the more the difference between the two interest rates adds up.
4. One worked example, then your turn
In the worked example, we find a forward exchange rate for the euro.
Now the foreign interest rate that a forward exchange rate implies, with hints, one step at a time.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for a forward exchange rate and one for an implied interest rate. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: