Forward Rate Calculator
From two spot rates of different maturities, derive the implied forward rate between those dates — the market's implied expectation of future rates.
Input Data
Results
At a glance:The forward rate is the implied rate for a future period (e.g. '1y1y') derived from two spot rates of different maturities, by no-arbitrage: (1 + r_far)^t_far = (1 + r_near)^t_near × (1 + f)^(t_far − t_near). Solving: f = [(1 + r_far)^t_far ÷ (1 + r_near)^t_near]^(1 ÷ (t_far − t_near)) − 1. An upward-sloping curve implies a forward rate above the short rate (expected rise). It is an implied expectation, not a precise forecast; returns 0 if far ≤ near.
Formula
f = [(1 + r_far)^t_far ÷ (1 + r_near)^t_near]^(1 ÷ (t_far − t_near)) − 1.
No-arbitrage: invest to far = invest to near then roll at forward.
Far years must exceed near years, else 0.
$$$(1+r_2)^{t_2} = (1+r_1)^{t_1}(1+f)^{t_2-t_1}$$$How to Use
- Enter the near (shorter) spot rate and its years.
- Enter the far (longer) spot rate and its years (greater than near).
- View the implied forward rate.
Implied forward rates from spot rates (annual compounding)
| Near spot | Near yrs | Far spot | Far yrs | Forward rate |
|---|---|---|---|---|
| 5% | 1 | 6% | 2 | 7.0095% |
| 4% | 1 | 4.5% | 2 | 5.0024% |
| 3% | 2 | 4% | 3 | 6.0292% |
Case Studies
Case 1: Derive the forward rate by no-arbitrage
Market: 1y spot r1 = 5%, 2y spot r2 = 6%. Find the 1y1y forward f.
No-arbitrage: (1.06)^2 = (1.05)^1 × (1 + f) → 1.1236 = 1.05 × (1 + f) → 1 + f = 1.070095 → f ≈ 7.0095%.
Interpretation: locking 2 years at 6% gives 1.1236 per dollar; to match by '1y at 5% then roll 1y', the second year must earn ~7.01%. This 7.01% is implied by the two spots, not quoted — if the real second-year rate deviates, arbitrage exists in theory.
Case 2: Forward rate reads the curve shape and expectation
Curve A upward: 1y 4%, 2y 4.5% → forward ≈ 5.00% (above both). Curve B inverted: 1y 5%, 2y 4% → forward below 4%.
Same two maturities: upward curve → high forward; inverted → low forward.
Interpretation: the forward is a magnifier of the curve. Upward slope implies the market expects rates to rise (expansion, inflation, hikes); inversion implies expected falls (slowdown/recession signal). But the forward embeds a term premium and is not a precise forecast — use it to understand shape and for relative pricing, not as a crystal ball. Annual compounding assumed; far years must exceed near.
FAQ
What is a forward rate vs a spot rate?
The spot rate is locked today for a term (1-year, 2-year). The forward rate is for a future period ('1 year from now, for 1 year', 1y1y). It is not directly quoted but implied by spot rates via no-arbitrage: locking 2 years today should equal locking 1 year then rolling 1 year at the forward. Solve that equality for the implied forward rate — the market's implied expectation of that future period's rate.
How is it computed; can it predict future rates?
From no-arbitrage: (1 + r_far)^t_far = (1 + r_near)^t_near × (1 + f)^(t_far − t_near). Example: 1y 5%, 2y 6% → forward (1y1y) = [(1.06)^2 ÷ 1.05] − 1 ≈ 7.0095%. When the long rate (6%) exceeds the short (5%), the forward (7.01%) exceeds both — an upward-sloping curve implies expected rises. But the forward is NOT a precise forecast; it is the implied value under current spots and no-arbitrage, blended with a term premium. Actual rates depend on growth, inflation, policy and sentiment. Use it to read the curve shape and for pricing, not as a crystal ball.
What is it used for; cautions?
Uses: (1) bond and yield-curve analysis — derive forward rates to read steepness/inversion; (2) FRA and interest-rate-futures pricing; (3) interest-rate-swap valuation (float cash flows at forward rates); (4) rate-risk management. Cautions: far years must exceed near (else 0); use the same compounding (this uses annual); the result is implied, not guaranteed, and shifts with input spots; practical spots come from a zero curve. Educational/estimation only.
Relation among forward rate, forward premium, yield curve?
All three are about term structure. The yield curve plots spot rates by maturity (upward/flat/inverted). The forward rate is extracted from it — its slope determines the forward: steeper curve → higher implied forward; inverted → forward below the short rate. The forward premium/discount extends the same no-arbitrage logic from one currency's maturities (forward rate) to two currencies' rate gaps (FX). Same underlying logic, different dimension: curve is the raw data, forward rate its time extraction, forward premium its currency extension. Pair with the forward-premium and Fisher-equation calculators.
Why not a precise forecast of future rates?
The formula gives the implied expectation under current spots, but several gaps make it unreliable as a forecast: (1) term premium — the forward embeds expected rate + term premium (compensation for longer duration), usually biased high; (2) no-arbitrage friction — transaction costs, taxes, liquidity and market segmentation cause deviations; (3) uncertainty — future rates depend on unpredictable growth, inflation, policy, geopolitics. Historically forward rates systematically over-predict. So use it to read the curve shape, as a pricing benchmark (FRA, swaps) and for relative-value scenarios — not as the future rate.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.