Forward Rate Derivation from the Curve

Forward rates are the rates that make today's spot curve internally consistent. They are not forecasts. If a 5-year zero pays S₅ and a 2-year zero pays S₂, the 3-year rate locked in for the period from year 2 to year 5 — call it F₂,₅ — is fixed by no-arbitrage. Get F₂,₅ wrong by 5 basis points on a $100 million position and you have mispriced about $150,000 of carry. Get the inputs wrong — most often by reading par yields off Treasury H.15 and treating them as spot rates — and you have mispriced everything that follows.

This article does the math the right way: spot rates in, forward rates out, with the bootstrap step that turns par yields into spot rates made explicit.

The No-Arbitrage Identity

An investor with an n-year horizon has two strategies that must produce the same total return, or arbitrage exists:

• Strategy A: invest at the n-year spot rate Sₙ.
• Strategy B: invest at the m-year spot rate Sₘ, then roll into the forward rate Fₘ,ₙ for the remaining (n − m) years.

Setting the two terminal values equal:

(1 + Sₙ)ⁿ = (1 + Sₘ)ᵐ × (1 + Fₘ,ₙ)⁽ⁿ⁻ᵐ⁾

Solving for the forward:

Fₘ,ₙ = [ (1 + Sₙ)ⁿ / (1 + Sₘ)ᵐ ]^(1/(n−m)) − 1

The derivation is standard fixed-income; Hull treats it in Options, Futures, and Other Derivatives (Ch. 4) and Fabozzi gives the same identity with bond-equivalent compounding conventions in Bond Markets, Analysis, and Strategies.¹ The expectations-hypothesis framing — that Fₘ,ₙ equals the market's expectation of the future spot rate — was Hicks (1939) and Lutz (1940). Empirical tests have rejected it for decades; forwards include a term premium, not just rate expectations (more on this below).²

The identity is unforgiving on one input: Sₙ and Sₘ must be spot (zero-coupon) rates. Substituting par yields invalidates the equation.

Where to Get Real Spot Rates

The Federal Reserve publishes a clean zero-coupon Treasury yield curve daily as a research dataset built on Gürkaynak, Sack, and Wright (2007).³ The series uses an extended Nelson-Siegel-Svensson fit to off-the-run coupon Treasuries; the SVENY columns are continuously compounded zero yields by maturity in years.

For the most recent data point at the time of writing (April 24, 2026):

Convert continuously compounded to annual with rₐ = e^(r_cc) − 1; the formulas above use annual compounding, so we will work from the right column.

Compare to the H.15 Selected Interest Rates release for April 27, 2026, which publishes constant-maturity Treasury yields:⁴

H.15 documents these as "yields on actively traded non-inflation-indexed issues adjusted to constant maturities" — interpolated par yields, not spot rates.⁴ In a near-flat curve like today's, par and spot agree to roughly 1–10 bp. In a steep curve with high coupons, they diverge enough to matter. Either way, the discipline is the same: bootstrap before you compute forwards.

Bootstrap: Par Yields → Spot Rates

The bootstrap is recursive. The 1-year point is a Treasury bill — already a zero — so S₁ equals the 1-year yield. Each longer maturity is solved by demanding that a par bond at the published par yield prices to 100 when discounted at the spot curve being constructed.

Take the H.15 figures above and assume annual coupons for clarity (Treasury notes actually pay semi-annual; convention adjustments below).

Step 1 — 1-year: S₁ = 3.69%.

Step 2 — 2-year: A 2-year par bond pays a coupon of 3.78 each year and 100 at maturity. Price = 100. Solve:

100 = 3.78 / (1 + S₁) + 103.78 / (1 + S₂)² 100 = 3.78 / 1.0369 + 103.78 / (1 + S₂)² 100 = 3.6454 + 103.78 / (1 + S₂)² 103.78 / (1 + S₂)² = 96.3546 (1 + S₂)² = 1.07707 S₂ = 3.781%

Par yield was 3.78%; spot is 3.78%. The divergence is 0.1 bp because the curve is essentially flat and coupons are low.

Step 3 — 3-year: Same logic, one more discount term:

100 = 3.83 / 1.0369 + 3.83 / (1.03781)² + 103.83 / (1 + S₃)³ 100 = 3.6938 + 3.5560 + 103.83 / (1 + S₃)³ (1 + S₃)³ = 103.83 / 92.7502 = 1.11949 S₃ = 3.832%

Again, ~0.2 bp drift from par. The honest practitioner takeaway: in this curve, you can use H.15 par yields as a working approximation for spots and your forward rates will be accurate to a basis point. The mechanic still must be understood, because curve regimes change.

To see the bootstrap actually bite, take a steeper hypothetical: S₁ = 2.00%, 2-year par yield = 5.00%.

100 = 5 / 1.02 + 105 / (1 + S₂)² 100 = 4.9020 + 105 / (1 + S₂)² (1 + S₂)² = 105 / 95.0980 = 1.10412 S₂ = 5.078%

Eight basis points above par — and that gap widens at longer maturities and higher coupons. A 30-year par bond on a steeply upward-sloping curve can sit 30–60 bp above the bootstrapped 30-year spot.⁵ Plug par into a forward formula in that environment and your numbers are wrong by a tradeable amount.

Compounding convention matters. Treasuries quote bond-equivalent (semi-annual) yields; H.15 follows that convention. To use the formulas above on bond-equivalent yields, either (a) convert to annual via (1 + y/2)² − 1, or (b) keep semi-annual compounding throughout: (1 + Sₙ/2)^(2n) replaces (1 + Sₙ)ⁿ. Pick a convention before you start; mixing produces nonsense.

Worked Example: F₂,₅ and F₅,₁₀ from GSW Spot Rates

Using the GSW spot yields above (annual compounding):

3-year forward, starting in 2 years (F₂,₅):

F₂,₅ = [(1.040159)⁵ / (1.038416)²]^(1/3) − 1 = [1.217577 / 1.078303]^(1/3) − 1 = [1.129163]^(1/3) − 1 = 1.041303 − 1 = 4.13%

Cross-check using continuously compounded GSW yields, where the forward identity collapses to a weighted difference: F₂,₅(cc) = (5 × 3.9373 − 2 × 3.7696) / 3 = 4.0492%. Convert to annual: e^0.040492 − 1 = 4.13%. ✓

5-year forward, starting in 5 years (F₅,₁₀):

F₅,₁₀ = [(1.045259)¹⁰ / (1.040159)⁵]^(1/5) − 1 = [1.556824 / 1.217577]^(1/5) − 1 = [1.278625]^(1/5) − 1 = 1.050379 − 1 = 5.04%

Both forwards sit above the 5-year spot (4.02%): the curve embeds rising rates at the long end, with the 5y5y forward priced about 102 bp above the current 5-year spot. Whether that pans out is a separate question.

Why Forwards Aren't Forecasts

In a frictionless world with risk-neutral investors, Fₘ,ₙ would equal the expected future spot rate. In the real world it doesn't, because investors demand extra yield to lock up money longer. That extra yield is the term premium. Gürkaynak, Sack, and Wright's data show the 10-year term premium has ranged from roughly −80 bp to +400 bp over the past four decades, with the post-2009 average closer to zero or slightly negative.³ The Adrian-Crump-Moench model, maintained by the New York Fed, gives daily term-premium estimates and historically explains a meaningful share of the variation in long-end forwards.⁶

What this means in practice:

• A 5y5y forward at 5.04% when the current 5-year spot is 4.02% does not imply the market thinks 5-year rates will be 5.04% in five years. It implies the market is willing to lock in 5.04% for that period — which is the expected future 5-year spot plus a term premium.
• Forwards have been systematically biased upward as forecasts when curves slope up. Backing out term premium gives a cleaner — though still imperfect — read on rate expectations.²⁶

Practical Uses

Breakeven for maturity-extension trades. You are choosing between rolling a 2-year note into a 3-year note versus buying a 5-year today. The 3-year forward starting in 2 years (F₂,₅) is 4.13%. If the 3-year spot in two years prints above 4.13%, the roll wins. Below, the 5-year wins. Forwards convert the maturity-choice question into a single rate to bet against.

Roll-down. A 5-year bond held for 1 year becomes a 4-year bond. If the curve shape holds, the bond reprices at the (lower) 4-year spot, generating a positive return on top of coupon. The forward curve tells you exactly how much roll-down is priced in. The 1-year forward starting in 4 years (F₄,₅) computed from GSW spots is the rate at which roll-down disappears.

Curve-shape implied views. Where the forward curve sits above the spot curve, the market embeds rising rates. Below, falling rates. Comparing the implied path to your own view (or to consensus surveys like the SPF) is the cleanest way to size relative-value positions.

Pitfalls

• Par yields read as spots. The default mistake. H.15 and the Treasury daily yield curve publish par yields. Use GSW for direct spots, or bootstrap.
• Compounding mismatches. Bond-equivalent in, annual out. Continuously compounded crossed with discrete. Convert before you compute.
• Notation drift. "2y3y" usually means a 2-year rate starting in 3 years (F₃,₅). Some desks invert it. Confirm convention before quoting a number.
• Forecast framing. A forward is the no-arbitrage rate, not the predicted rate. Strip the term premium before treating it as an expectation.

Next Step

Pull GSW spot yields for today's date.³ Compute three forwards: F₁,₂, F₂,₅, F₅,₁₀. Plot them next to the spot curve. Anywhere the forward sits more than ~50 bp from the corresponding spot is where the market is pricing in the strongest directional move — and where your view of rates either agrees with that pricing or trades against it.

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Related: Bootstrap a Spot Curve from Treasury Coupons · Term Premium Models in Practice · Carry and Roll-Down Decomposition

Footnotes

1. Hull, John C. Options, Futures, and Other Derivatives, 11th ed. (Pearson, 2022), Chapter 4, "Interest Rates," derives the no-arbitrage forward-rate identity. Fabozzi, Frank J. Bond Markets, Analysis, and Strategies, 10th ed. (Pearson, 2021), Chapter 5, presents the same derivation with bond-equivalent yield conventions used in U.S. Treasury markets. ↩

2. Hicks, John R. Value and Capital (Oxford University Press, 1939) and Lutz, Friedrich A., "The Structure of Interest Rates," Quarterly Journal of Economics 55, no. 1 (1940): 36–63, are the canonical statements of the expectations hypothesis. Modern empirical tests almost universally reject the pure expectations hypothesis in favor of a time-varying term-premium specification; see Cochrane, John H., and Monika Piazzesi, "Bond Risk Premia," American Economic Review 95, no. 1 (2005): 138–160. ↩ ↩²

3. Gürkaynak, Refet S., Brian Sack, and Jonathan H. Wright, "The U.S. Treasury Yield Curve: 1961 to the Present," Journal of Monetary Economics 54, no. 8 (2007): 2291–2304. Daily updated zero-coupon yield curve data is published by the Federal Reserve Board at https://www.federalreserve.gov/data/nominal-yield-curve.htm; download the CSV at https://www.federalreserve.gov/data/yield-curve-tables/feds200628.csv. The SVENYxx columns are continuously compounded zero yields at year xx; the SVPYxx columns are par yields. ↩ ↩² ↩³

4. Federal Reserve, Selected Interest Rates (Daily) — H.15, https://www.federalreserve.gov/releases/h15/. Constant-maturity Treasury yields are interpolated by the U.S. Treasury from the daily yield curve for non-inflation-indexed Treasury securities. They are par yields, not spot rates. Methodology: U.S. Treasury, Treasury Yield Curve Methodology, https://home.treasury.gov/policy-issues/financing-the-government/interest-rate-statistics/treasury-yield-curve-methodology. ↩ ↩²

5. For a worked walkthrough showing how par-spot divergence widens with curve slope and maturity, see CFA Institute, Fixed Income: Yield Curves and the Bootstrapping Method, in the Level II curriculum readings on fixed-income valuation. ↩

6. Adrian, Tobias, Richard K. Crump, and Emanuel Moench, "Pricing the Term Structure with Linear Regressions," Journal of Financial Economics 110, no. 1 (2013): 110–138. Daily term-premium estimates: Federal Reserve Bank of New York, ACM Term Premium, https://www.newyorkfed.org/research/data_indicators/term-premia-tabs. ↩ ↩²