Why is the forward rate arithmetic, not a forecast?
Ask ten people what a one-year forward exchange rate means and most will say "where the market thinks the currency will be." That answer is wrong, and being clear about why is the foundation of everything else in this unit.
Two ways to own dollars in ninety days
Suppose EUR/USD is 1.0800 dollars per euro, the 90-day dollar rate is 4.50% and the 90-day euro rate is 2.50%. You have $1,080,000 and you want dollars in 90 days. Two routes:
Route A — stay in dollars. Deposit at 4.50% on an actual/360 basis:
1,080,000 × (1 + 0.045 × 90/360) = 1,080,000 × 1.011250 = $1,092,150
Route B — go via euros, hedged. Convert at 1.0800 to get €1,000,000, deposit at 2.50%:
1,000,000 × (1 + 0.025 × 90/360) = €1,006,250
and sell that euro amount forward for dollars today, at a rate F agreed now.
Both routes are riskless and start with the same money. If they ended with different amounts, anyone could borrow in the cheap route and lend in the rich one, riskless, in unlimited size. So they must end equal:
F = 1,092,150 / 1,006,250 = 1.0854
The formula
Generalised, for a pair quoted as units of the terms currency per one unit of the base currency:
F = S × (1 + i_terms × t) / (1 + i_base × t)
Check it: 1.0800 × 1.011250 / 1.006250 = 1.0854. The 54-pip difference from spot is the forward points, and the annualised premium is (1.085366 / 1.0800 − 1) × 4 = 1.99% ≈ the 2.00 point interest differential. Work it from the displayed 1.0854 instead and you get exactly 2.00%: the missing basis point is the rounding, not a gap in the parity.
This is covered interest parity (CIP). Note the direction, because it catches everyone: the currency with the higher interest rate — here the dollar — trades at a forward discount. Its rate advantage is given straight back in the forward price.
What happens if the forward is wrong
Say the market quoted 1.0900 instead. Sell the €1,006,250 forward at 1.0900 and you receive $1,096,812.50 against $1,092,150 from the dollar deposit — a riskless $4,662.50 on the same starting capital. In a market turning over trillions a day, that gap is arbitraged away in seconds. CIP holds not because it is true but because violating it is a free lunch.
The link back to the carry trade
Now redo the previous unit. AUD/JPY spot 100.00, one-year Australian rate 4.50%, Japanese rate 0.50%:
F = 100.00 × 1.005 / 1.045 = 96.17
That is the exact breakeven you computed for the carry trade. It has to be. The forward rate is the level at which the carry trade earns nothing — which means a carry trade that hedges its currency risk in the forward market earns precisely zero, before costs. The carry trade is unhedged by construction. There is no version of it that keeps the interest and removes the risk.
Where the arithmetic leaks
CIP is an arbitrage condition, so it holds as tightly as the arbitrage is cheap to run. Since 2008 it has not held exactly. Balance-sheet and regulatory capital costs make the trade expensive for the banks that used to run it, and demand for dollar funding is persistently one-sided — so a small, stubborn cross-currency basis remains, typically a few to a few tens of basis points for major pairs, and it widens reliably at quarter- and year-ends when bank balance sheets are reported. Du, Tepper and Verdelhan documented this in 2018. The point is not that CIP is broken; it is that "riskless arbitrage" costs balance sheet, and balance sheet is no longer free.
The sentence to keep
A forward rate is a spot rate adjusted so that nobody gets paid twice. It contains no opinion about the future. If you read it as a forecast, you are reading the interest differential and calling it a prediction — and the next lesson is about what happens when people take that prediction seriously.
In the data
Every input to the parity arithmetic is public and the output is not. Spot and the two overnight rates are below, today's euro-dollar and the benchmarks for each currency.
A quoted forward is a dealer's price in an over-the-counter market, and no free public feed carries one. You can compute the theoretical forward from these three numbers to as many decimals as you like and never once compare it against a traded one, which makes any deviation from parity unobservable from here rather than absent from the market.
Try it now
- The first table is euro-dollar's close on four August dates a year apart. Under it are the Fed's and the ECB's policy rates on each of the first three. Start with 1 August 2025: its close is your spot, the top of the Fed's band your dollar rate and the ECB deposit rate your euro rate. Policy rates are standing in for one-year deposit rates here, so write down which rates you used. The year since is on the chart; drop a Level at that starting rate.
- Compute the implied one-year forward rate with the formula above. The euro is the base currency and the dollar the terms currency.
- The spot a year later is the next row of the first table. How far is the actual outcome from the forward you just computed? Repeat from 1 August 2023 and from 1 August 2024, each with its own rates, and write one sentence on what the three gaps demonstrate about forwards as forecasts.