Forward Rates for Multiple Years
To get a forward exchange rate more than one year out, you compound the interest (or inflation) ratio for each year instead of using it once. It is the same parity idea, just raised to the power of n. The gap between the two currencies widens the further out you go.
What you will learn
- How to extend the one-year forward formula to n years
- Why you raise the rate ratio to the power of n
- How to build a year-by-year table of forward rates
- Using the same method with inflation (PPP) forecasts
- Common mistake: multiplying the rate gap by n instead of compounding
The formula
- S₀
- spot rate, in units of currency A per 1 unit of currency B
- Fₙ
- forward rate for delivery in n years, same quote
- r_A, r_B
- annual interest rates in currencies A and B
- n
- number of years
Worked example
The spot rate is ¥150 per $1. Annual interest rates are 1% in Japan and 4% in the US. Find the forward rates for years 1 to 3.
- Quote is yen per dollar, so A = yen and B = dollar.
- Yearly ratio = 1.01 ÷ 1.04 = 0.97115.
- F₁ = 150 × 0.97115 = ¥145.67.
- F₂ = 150 × 0.97115² = ¥141.47.
- F₃ = 150 × 0.97115³ = ¥137.39.
Answer: The three-year forward rate is about ¥137.39 per dollar. The dollar trades at a growing forward discount because US rates are higher each year.
Common questions
How do you calculate a forward exchange rate for 2 or 3 years?
Take the one-year parity ratio, (1 + r_A) ÷ (1 + r_B), raise it to the number of years, and multiply by today's spot rate. For two years use the ratio squared, for three years cubed. Make sure both interest rates are annual rates for that horizon.
Why do you compound instead of multiplying by the number of years?
Interest compounds, so money invested in each currency grows by (1 + r) every year. The forward rate has to match those compounded balances. Simply multiplying the rate gap by n ignores interest on interest and gets more wrong the longer the horizon.
Can you use inflation instead of interest rates?
Yes. Relative purchasing power parity works the same way: E(Sₙ) = S₀ × [(1 + i_A) ÷ (1 + i_B)]ⁿ using expected inflation rates. This gives a forecast of the future spot rate rather than a no-arbitrage forward price.
Are long-dated forward exchange rates accurate forecasts?
They are the arbitrage-free price for locking in a rate today, not a promise of where the spot rate will be. Actual exchange rates can drift far from them over several years, which is exactly why companies use forwards to hedge long-term foreign currency cash flows.