Forward Rate Calculator

Derive the forward interest rate implied for a future period from two spot (zero-coupon) rates and their maturities, using the standard no-arbitrage bootstrapping formula.

Quick Facts

Formula
F = [(1+S2)^T2 / (1+S1)^T1]^(1/(T2-T1)) - 1
S1, S2 are spot rates for the near and far maturities T1, T2; F is the annualized forward rate for the period between them.
No-arbitrage logic
Near rate then forward rate = far rate
$1 invested at S1 for T1 years, then rolled into F for the rest, grows to the same value as $1 invested directly at S2 for T2 years.

Your Results

Calculated
Implied forward rate
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Annualized rate for the period between T1 and T2
Forward period length
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Time span the forward rate covers (T2 - T1)
Total growth over period
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Compounded return over the forward period, not annualized
Future value of $1
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Same whether compounded at the far spot rate or rolled via near + forward

Ready

Enter both spot rates and maturities, then press Calculate.

How the Forward Rate Calculator works

A forward rate is the interest rate implied by today's yield curve for a loan or investment that starts at some point in the future. If you know the spot (zero-coupon) rate for a near maturity and for a farther maturity, the market's current pricing already tells you what rate must apply to the period between those two dates — otherwise there would be an arbitrage opportunity. This calculator solves for that implied rate.

The formula

Given a near-period spot rate S1 for T1 years and a far-period spot rate S2 for T2 years (T2 > T1), the annualized forward rate for the period between T1 and T2 is:

F = [(1 + S2)T2 / (1 + S1)T1]1/(T2−T1) − 1

The logic is no-arbitrage: $1 invested at the near spot rate for T1 years and then rolled into the forward rate for the remaining (T2 − T1) years must grow to exactly the same amount as $1 invested directly at the far spot rate for T2 years. Solving that equation for F gives the formula above.

Worked example

Suppose the 1-year spot rate is 4.0% and the 5-year spot rate is 4.6%. With T1 = 1, S1 = 4.0%, T2 = 5, S2 = 4.6%, the forward factor works out to 1.0465 / 1.041 ≈ 1.2040, and raising that to the power 1/(5 − 1) gives an implied forward rate of about 4.75% per year for the period from year 1 to year 5. Because that forward rate sits above both the 1-year and 5-year spot rates, the yield curve is getting steeper further out.

Reading the curve shape from the forward rate

  • Forward rate above both spot rates: the curve is upward-sloping and getting steeper between the two maturities — longer-dated money is priced increasingly higher.
  • Forward rate between the two spot rates: a typical, gently upward-sloping curve where the marginal rate sits between the near and far yields.
  • Forward rate below both spot rates: the curve is inverted between these maturities, often read as a sign that rates or growth expectations are cooling for that stretch.

What this calculator assumes

The formula assumes annual compounding and that both spot rates are quoted on the same compounding basis (both zero-coupon, or both bond-equivalent yields). It computes the theoretical, no-arbitrage forward rate implied by the two spot rates you enter — not a dealer-quoted forward rate agreement (FRA) price, which can include additional credit and liquidity premia. Use it to understand what a yield curve implies, not as a live tradable market quote.

Frequently Asked Questions

What is a forward rate?
A forward rate is the interest rate that today's spot (zero-coupon) yield curve implies for a loan or investment beginning at a future date. It is not a prediction of future rates — it is the break-even rate that keeps a two-step investment (near spot rate, then forward rate) growing to the same value as a one-step investment at the far spot rate.
How do you calculate the forward rate from two spot rates?
Use F = [(1+S2)^T2 / (1+S1)^T1]^(1/(T2-T1)) - 1, where S1 and S2 are the spot rates for the near maturity T1 and far maturity T2. Raise each spot rate's compounding factor to its maturity, divide the far factor by the near factor, then take the root of the remaining period length (T2-T1).
Why is the forward rate sometimes higher than both spot rates?
The far-maturity spot rate S2 is effectively a blended average of the near-period rate and the forward rate covering the remaining years. If the yield curve is getting steeper as maturity increases, the marginal forward rate for that far segment has to run above the blended average — which is itself above the near rate — to pull the average up to S2.
Is this the same as a Forward Rate Agreement (FRA) quote?
No. This calculator gives the theoretical forward rate implied purely by the two spot rates you enter, assuming no arbitrage. A traded FRA or bank-quoted forward rate can differ slightly because it also prices in credit risk, liquidity, and dealer margin on top of the pure yield-curve math.

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