Forward Rate Derivation from the Curve

Forward rates are implied future interest rates extracted from today's spot curve. They answer a specific question: "What short-term rate does the current yield curve embed for a future period?" The math is straightforward once you understand that spot curves must be consistent with each other or arbitrage opportunities would exist. The durable lesson: forward rates connect two points on the spot curve through a no-arbitrage relationship (where investing for the full period must equal investing short-term and rolling over at the implied forward rate).

The Core Formula (Why It Works)

The forward rate formula derives from a simple principle: an investor should be indifferent between two equivalent strategies.

Strategy A: Invest for n years at the n-year spot rate.

Strategy B: Invest for m years at the m-year spot rate, then reinvest for (n-m) years at the forward rate from year m to year n.

If these strategies produced different returns, you could borrow using one and lend using the other for riskless profit. Markets eliminate this arbitrage, so:

(1 + Sn)^n = (1 + Sm)^m x (1 + Fm,n)^(n-m)

Where:

• Sn = spot rate for n years
• Sm = spot rate for m years
• Fm,n = forward rate from year m to year n

Solving for the forward rate:

Fm,n = ((1 + Sn)^n / (1 + Sm)^m)^(1/(n-m)) - 1

The point is: forward rates are not forecasts. They're the rates that make today's spot curve internally consistent. Whether future rates actually match implied forwards is an entirely separate question.

Step-by-Step Calculation: The 1-Year Forward Rate, 2 Years from Now

Setup: You observe spot rates from the Treasury curve:

• 2-year spot rate: 3.59%
• 3-year spot rate: 3.48%

You want the 1-year forward rate starting in year 2 (written as F2,3 or "1y2y" in practitioner notation).

Calculation:

F2,3 = ((1.0348)^3 / (1.0359)^2)^(1/1) - 1

F2,3 = (1.1078 / 1.0731)^1 - 1

F2,3 = 1.0323 - 1

F2,3 = 3.23%

Interpretation: The market embeds a 3.23% rate for 1-year money starting 2 years from now. This is lower than both current spot rates, which makes sense given the curve's shape at these maturities (the 3-year yield is lower than the 2-year yield, indicating expectations of declining rates).

Reading the December 2025 Treasury Curve for Forward Rate Insights

Current spot rates (approximately equal to par yields in the near-flat environment) as of December 2025 (Source: Federal Reserve H.15):

Forward rate from year 1 to year 2 (F1,2):

F1,2 = ((1.0359)^2 / (1.0366)^1)^(1/1) - 1 = 3.52%

Forward rate from year 2 to year 5 (F2,5), a 3-year rate starting in 2 years:

F2,5 = ((1.0345)^5 / (1.0359)^2)^(1/3) - 1

F2,5 = (1.1854 / 1.0731)^(0.333) - 1

F2,5 = 3.36%

Forward rate from year 5 to year 10 (F5,10):

F5,10 = ((1.0367)^10 / (1.0345)^5)^(1/5) - 1

F5,10 = (1.4334 / 1.1854)^(0.20) - 1

F5,10 = 3.89%

Why this matters: The 5-year forward rate starting in 5 years (3.89%) exceeds the current 5-year spot rate (3.45%) by 44 bps. The curve implies rates will rise at the long end. Whether that happens depends on future Fed policy, inflation, and term premiums (all unknowns).

What Forward Rates Actually Tell You (And What They Do Not)

Forward rates reveal:

• Market-implied reinvestment rates for future periods
• Breakeven rates for comparing different maturity strategies
• The shape consistency embedded in today's curve
• Relative value signals when forwards seem extreme versus expectations

Forward rates do not reliably predict:

• Actual future spot rates (forward rates have historically been poor forecasters)
• Fed policy decisions (though they incorporate some expectations)
• Where rates "should" go (the term premium distorts the signal)

Research shows that forward rates consistently overestimate future short rates when curves slope upward (Gurkaynak, Sack, and Wright, 2006). This reflects a term premium (investors demand extra yield for locking up money longer), not purely rational expectations of rate increases.

Using Forward Rates for Investment Decisions

The Breakeven Framework

Forward rates set the breakeven hurdle for maturity extension decisions.

Example: You're choosing between a 2-year Treasury at 3.59% and a 5-year Treasury at 3.45%.

The 3-year forward rate from year 2 to 5 (F2,5) is 3.36%. This means:

• If 3-year rates in 2 years are above 3.36%, you would have been better off buying the 2-year and rolling into the 3-year later
• If 3-year rates in 2 years are below 3.36%, buying the 5-year today wins

The forward rate is your decision threshold. You're not betting on whether rates go up or down in absolute terms (you're betting on whether they exceed or fall short of the implied forward).

Roll-Down Analysis

Forward rates also drive roll-down return calculations. When the curve slopes upward, a bond "rolls down" to lower yields as it ages (assuming the curve shape holds constant). The magnitude of roll-down depends on the forward rate structure.

If the 5-year forward rate 1 year from now is lower than the current 5-year rate, a 5-year bond held for 1 year will benefit from both carry (coupon income) and price appreciation (as it becomes a 4-year bond priced at a lower yield).

Detection Signals: You're Likely Calculating Forwards Incorrectly If...

• Your forward rate is negative when both spot rates are positive (check your formula order)
• Your forward rate exceeds both spot rates in a normal upward-sloping curve (mathematically impossible)
• You're deriving forwards from par yields instead of spot rates (bootstrap first, then calculate forwards)
• Your implied forwards seem radically disconnected from the spot curve shape (arithmetic error probable)

Common Mistakes and How to Avoid Them

Mistake 1: Using par yields directly

Forward rates derive from spot rates, not par yields. Using par yields produces forward rates that don't satisfy no-arbitrage conditions. Bootstrap your spot curve first (Analyst Prep, 2024).

Mistake 2: Confusing notation

"3y2y" means the 3-year rate starting 2 years from now. "2y3y" means the 2-year rate starting 3 years from now. These are different rates. Confirm your notation convention.

Mistake 3: Treating forwards as forecasts

The 10-year forward rate 5 years from now is not the market's best guess of where the 10-year rate will be in 5 years. It's the rate that makes the current 5-year and 15-year spot rates internally consistent. Big difference.

Mistake 4: Ignoring compounding conventions

If your spot rates are semi-annual bond equivalent yields, convert them to the appropriate compounding basis before applying the forward formula. Mixing conventions produces nonsensical results.

Checklist: Forward Rate Derivation

Essential (get these right first)

• Start from spot rates, not par yields (bootstrap if necessary)
• Verify your answer makes sense relative to the spot curve shape
• Match compounding conventions across all rates in the calculation

High-impact refinements

• Calculate breakeven rates for specific investment decisions
• Use forward curves to identify when the market prices in unusually large or small rate moves
• Combine forward analysis with key rate duration to understand curve reshaping risks

Practical Exercise: Build a Forward Rate Table

Using current Treasury spot rates, calculate:

1. The 1-year forward rate 1 year from now (F1,2)
2. The 1-year forward rate 2 years from now (F2,3)
3. The 2-year forward rate 3 years from now (F3,5)
4. The 5-year forward rate 5 years from now (F5,10)

Plot these against the current spot curve. Where the forward curve lies above the spot curve, the market implies rising rates. Where it lies below, falling rates. Compare to your own rate expectations to identify potential mispricings (remembering that forwards include term premiums, not just rate expectations).

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Related: Spot Curves vs. Par Curves | Understanding Treasury Yield Curve Shapes | Yield to Call and Yield to Worst