How two spot rates imply a single forward rate
The yield curve gives you spot rates — one rate for each maturity, each describing a single investment from today to that date. A forward rate takes two of those points and asks a sharper question: what rate must apply between them so that no riskless profit is possible? Investing once for the full long period has to equal investing for the shorter period and then reinvesting at the forward rate for the remaining time. Solving that equality for the unknown forward rate is all the formula below does; it is pure no-arbitrage, not a forecast. This complements abond pricing & YTM calculator, where those same spot rates are used to discount cash flows.
The forward rate formula
f = [ (1 + r₂)^t₂ ÷ (1 + r₁)^t₁ ]^(1 / (t₂ − t₁)) − 1
where r₁ and t₁ are the near (shorter) spot rate and its maturity, and r₂ and t₂ are the far (longer) spot rate and its maturity. The result f is the implied rate for the period from t₁ to t₂ — the rate at which rolling the short investment forward exactly matches holding the long one.
What makes this calculator different
- Works for any two maturities. Plug in any near and far spot rate with their times — not just the textbook one-year-in-one-year case — and it returns the forward rate for the period between them.
- It explains the no-arbitrage logic. The result is framed as the break-even rate that makes a long investment equal a short one rolled forward, so you see why the number is what it is, not just the number.
- It distinguishes forward from forecast. The forward rate is the market’s implied break-even, not a prediction of where rates will actually go — a distinction this tool keeps front and centre.
Frequently asked questions
What is a forward rate?+
A forward rate is the interest rate the market implies for a future borrowing or lending period, locked in today. It is not quoted directly — it is extracted from two spot rates on the yield curve by no-arbitrage. Given a near rate r₁ to maturity t₁ and a far rate r₂ to maturity t₂, the forward rate f for the period between them is f = [ (1 + r₂)^t₂ ÷ (1 + r₁)^t₁ ]^(1 / (t₂ − t₁)) − 1. In words, it is the rate that makes a long investment equal to a short one rolled into the forward period.
How does a forward rate differ from a spot rate?+
A spot rate is today’s rate for a single maturity: lend money now, get it back with interest at one future date. A forward rate is the rate implied between two future dates — for example, the one-year rate starting one year from now. Spot rates are the raw points on the yield curve; forward rates are derived from them. The whole forward curve is just a different way of expressing the same information the spot curve already contains.
Why is the forward rate set by no-arbitrage?+
Because two strategies must produce the same result. You can invest for the full long period at the far spot rate, or invest for the short period at the near spot rate and reinvest the proceeds at the forward rate for the remaining time. If those two paths gave different returns, a trader could borrow at the cheaper one and lend at the dearer one for a riskless profit. The forward rate is precisely the value that closes that gap, so it is fixed by arbitrage rather than by anyone’s opinion.
Is the forward rate a forecast of future rates?+
No. It is the market’s break-even rate, not a prediction. The forward rate tells you the future rate that would make holding a long bond equivalent to rolling short ones — nothing more. Actual future spot rates routinely differ from today’s forwards, partly because of risk and term premia. Treat the forward rate as a no-arbitrage benchmark, not as a crystal ball.
What is the forward rate used for?+
Forward rates are the building blocks for pricing forward rate agreements (FRAs), interest rate swaps, and other rate derivatives, where future floating payments are valued off the implied forward curve. They also let you read the market’s rate expectations straight from the shape of the yield curve — an upward-sloping curve implies forwards above current spot rates. Anyone hedging or valuing future cash flows leans on forward rates to do it consistently.
Disclaimer: This calculator is foreducation and illustration only. Forward rates derived here are no-arbitrage implications of the spot rates you enter, not forecasts of future interest rates, and real market quotes will differ with conventions, compounding, and risk premia. Nothing here is investment, tax, or trading advice.