DP
Debt & Yield Curvesintermediate

Forward Interest Rates

A forward interest rate is the rate implied by today's zero curve for borrowing or lending over a future period [T1,T2][T_1, T_2]. It is the rate that makes investing in a zero-coupon bond to T2T_2 equivalent to a zero to T1T_1 rolled into a forward zero from T1T_1 to T2T_2, by no-arbitrage. Forward rates are inputs for forward rate agreements (FRAs), interest rate futures, swaps, and the valuation of any contract whose cash flows reset against future short rates.

Why it matters

If 1-year zero is $3\%$ and 2-year zero is $4\%,themarkethasimplicitlypricedwhatratemustapplyinyear2forthetwostrategiestobreakeven.Thatimplicitrateisthe1āˆ’yearforwardstartingin1year.Itisāˆ—āˆ—notaforecastāˆ—āˆ—ofwhattheratewillactuallybe.Itistheāˆ—āˆ—breakāˆ’evenrateāˆ—āˆ—thatpreventsarbitragebetweeninvestingfortwoyearsat, the market has implicitly priced what rate must apply in year 2 for the two strategies to break even. That implicit rate is the 1-year forward starting in 1 year. It is **not a forecast** of what the rate will actually be. It is the **break-even rate** that prevents arbitrage between investing for two years at 4\%$ versus one year at $3\%$ and rolling into a one-year forward. Realised future spot rates routinely differ from forward rates by a term premium.

Formulas

Forward rate (continuous compounding)
fT1,T2=r2T2āˆ’r1T1T2āˆ’T1f_{T_1, T_2} = \frac{r_2 T_2 - r_1 T_1}{T_2 - T_1}
Hull (2022) eq. 4.5. Continuous compounding, zero rates r1r_1 and r2r_2 for maturities T1T_1 and T2T_2.
Forward rate (annual compounding)
(1+fT1,T2)T2āˆ’T1=(1+r2)T2(1+r1)T1(1 + f_{T_1, T_2})^{T_2 - T_1} = \frac{(1 + r_2)^{T_2}}{(1 + r_1)^{T_1}}
Algebraically equivalent under annual compounding. For T1=1T_1 = 1 and T2=2T_2 = 2 this collapses to (1+r2)2=(1+r1)(1+f1,2)(1 + r_2)^2 = (1 + r_1)(1 + f_{1,2}).

Worked examples

Scenario

1-year zero rate r1=3.00%r_1 = 3.00\%, 2-year zero rate r2=3.50%r_2 = 3.50\%, continuous compounding. Compute the 1-year forward rate starting in 1 year.

Solution

f1,2=(0.035Ɨ2āˆ’0.030Ɨ1)/(2āˆ’1)=(0.070āˆ’0.030)/1=0.040=4.00%f_{1,2} = (0.035 \times 2 - 0.030 \times 1)/(2 - 1) = (0.070 - 0.030)/1 = 0.040 = 4.00\%. The market implies a continuously compounded $4.00\%$ rate for year 2.

Scenario

Annual-compounding example. 1-year zero r1=4.00%r_1 = 4.00\%, 2-year zero r2=5.00%r_2 = 5.00\%.

Solution

(1+f1,2)=(1.05)2/1.04=1.1025/1.04=1.06010(1 + f_{1,2}) = (1.05)^2/1.04 = 1.1025/1.04 = 1.06010. Then f1,2=6.01%f_{1,2} = 6.01\%, consistent with the Hull continuous formula giving (2Ɨ0.05āˆ’1Ɨ0.04)=0.06(2 \times 0.05 - 1 \times 0.04) = 0.06 (approximate because continuous and annual differ by a small convexity).

Common mistakes

  • āœ—Forward rates are forecasts of future spot rates. They are no-arbitrage implied rates, not predictions. Under the expectations hypothesis they would equal expected spot rates, but empirical studies (Fama and Bliss 1987, Campbell and Shiller 1991) reject this. Realised future spots typically differ by a term premium.
  • āœ—Forward rates apply only to government securities. Forward rates exist for any zero curve. OIS forward rates anchor swap pricing, BBSW forward rates drive AUD interest rate swap fixings, and SOFR forward rates underpin USD post-LIBOR swap markets.
  • āœ—If forward rates rise, future spot rates will rise. Forward rates depend on the current curve shape. A steep curve mechanically produces high forwards even if the central bank is on hold. Inferring future policy directly from forwards requires stripping out the term premium, which is the focus of models like ACM (Adrian, Crump, Moench 2013).

Revision bullets

  • ā€¢Implied rate for a future period from today's zero curve
  • ā€¢Derived by no-arbitrage between two zero-coupon strategies
  • ā€¢f=(r2T2āˆ’r1T1)/(T2āˆ’T1)f = (r_2 T_2 - r_1 T_1)/(T_2 - T_1) continuous compounding
  • ā€¢Inputs into FRAs, swaps, interest rate futures
  • ā€¢Not a forecast, the break-even rate
  • ā€¢Differ from realised spots by a term premium

Quick check

Given a 1-year zero rate of $4\%$ and a 2-year zero rate of $5\%$ (continuous compounding), the 1-year forward rate starting in 1 year is

Why are forward rates not equal to expected future spot rates in practice?

Connected topics

FRAZero Rates

In learning paths

Fixed Income & Rates Path

Sources

  1. Hull (2022), Ā§4.7
    Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
    Derives the continuous-compounding forward rate formula and explains the no-arbitrage interpretation.
  2. Fama and Bliss (1987)
    Fama, Eugene F. and Robert R. Bliss. The Information in Long-Maturity Forward Rates. American Economic Review, Vol. 77, No. 4, 1987.
    Foundational empirical paper showing that forward rates contain information about expected returns rather than only expected future spot rates.
  3. Adrian, Crump and Moench (2013)
    Adrian, Tobias, Richard K. Crump, and Emanuel Moench. Pricing the Term Structure with Linear Regressions. Journal of Financial Economics, Vol. 110, 2013.
    NY Fed term-premium model now used by central bank researchers to separate forwards into expected rates and risk compensation.
  4. Fabozzi (2021), Ch. 5
    Fabozzi, Frank J. Bond Markets, Analysis, and Strategies. 10th ed. MIT Press, 2021. ISBN 978-0-262-04627-3.
    Comprehensive treatment of forward rates, their derivation from zeros, and their use across fixed-income products.
  5. RBA Statistical Table F17
    Reserve Bank of Australia. Statistical Tables F17 Zero-Coupon and Forward Interest Rates. RBA, monthly release.
    Publishes Australian zero-coupon and implied forward rate curves used for AUD interest rate analysis.
How to cite this page
Dr. Phil's Quant Lab. (2026). Forward Interest Rates. Derivatives Atlas. https://phucnguyenvan.com/concept/forward-interest-rates
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