Forward Rate
Implied future interest rate for the period between two tenors, derived from spot rates via no-arbitrage. The forward rate is what makes "invest at the near spot then roll" produce the same return as "invest at the far spot directly." Both spot inputs and the forward output share the same compounding convention (m), so semi-annual Treasury spots produce a semi-annual forward, annually-compounded spots produce an annual forward, and so on.
When to use: Use to extract market-implied future rates from the current term structure, or to compare your own rate forecast to what the market is already pricing in. A central input to swap valuation and rate-strategy trading. Set m to match the convention of the spot rates you are feeding in (m = 2 for US Treasuries, m = 1 for effective annual, m = 12 for monthly).
Formula
Variables
| Symbol | Name | Description | Unit |
|---|---|---|---|
| Forward | Forward Rate | Implied forward rate for the period between the two tenors | % |
| Spot1 | Near Spot Rate | Spot rate to the near tenor, as a decimal | % |
| Years1 | Near Tenor | Years to the near tenor | years |
| Spot2 | Far Spot Rate | Spot rate to the far tenor, as a decimal | % |
| Years2 | Far Tenor | Years to the far tenor | years |
| m | Coupons per Year | Number of coupons paid per year (e.g. 2 for semi-annual) | integer |
Real-Life Examples
Example 1: 1y1y Forward from 1y and 2y Treasury Spots (m = 2)
1-year spot = 4% nominal semi-annual, 2-year spot = 5% nominal semi-annual. What is the 1-year rate, 1 year from now (the "1y1y forward")?
Given
Step-by-Step
The market implies that 1-year rates one year out will be ~6.01% (nominal semi-annual). If you expect 1y rates a year out to be lower than 6%, lock in the 2-year spot now; if you expect higher, roll the 1-year. Use m = 1 instead if your spots are quoted as effective annual rates.
Example 2: Effective-Annual Forward (m = 1)
Same numerical 1y and 2y spots (4% and 5%), now interpreted as effective annual rates.
Given
Step-by-Step
When spots are quoted on an effective-annual basis (m = 1), the periodic-compounding form collapses to the textbook (1 + s2)^t2 / (1 + s1)^t1 expression — the forward is the geometric difference. Same numerical answer; different compounding convention.
Frequently Asked Questions
Because spot rates carry a compounding convention. US Treasury yields are quoted nominal semi-annual (m = 2); European bonds are often effective annual (m = 1); some money-market rates are monthly (m = 12). The forward-rate formula is exact only when both spots and the forward share the same convention — m makes that explicit so the computed forward comes out in the same convention you fed in.
Not exactly — it's an arbitrage-free implication of current spot rates. If the spot curve is upward-sloping, forward rates are higher than current spot rates regardless of expectations. The "expectations hypothesis" claims forwards equal expected future rates, but term-premium effects mean forwards usually overstate expected future rates.
A futures rate (like SOFR futures) is observable in market prices and includes convexity adjustments. The mathematical forward rate from the spot curve is a clean no-arbitrage figure; the two converge but differ in practice for long tenors.
Yes — when the spot curve is steeply inverted, forward rates implied by adjacent tenors can be negative. Common in some European yield curves during ECB negative-rate regimes.