Enter Values

years
Starting period for forward rate
years
Ending period for forward rate
%
t1-year spot rate
%
t2-year spot rate
%
Forward rate from t1 to t2

Notation Reference

r1Spot rate for period t1
r2Spot rate for period t2
t1Shorter time period
t2Longer time period
fForward rate from t1 to t2
Ryan O'Connell, CFA
Calculator by Ryan O'Connell, CFA

Calculation Result

Implied Forward Rate 6.01% Normal Typical forward rate level
Growth Factor (t1) 1.0400
Growth Factor (t2) 1.1025
Forward Period 1.00 yrs
Forward Growth 1.0601

Formula Breakdown

Forward Rate: (1 + r1)t1 * (1 + f)(t2-t1) = (1 + r2)t2
Rearranged to solve for: f = [(1+r2)t2 / (1+r1)t1]1/(t2-t1) - 1

Forward Rate Interpretation

Typical Pattern Yield Curve Shape Common Interpretation
f > r2 > r1 Upward Sloping Rising rates or term premium
f ≈ r2 ≈ r1 Flat Stable outlook
f < r2 < r1 Inverted Falling rates expected
Note: These patterns are typical but not universal. Forward rates embed both expectations and risk premiums.

Understanding Forward Rates

What is a Forward Rate?

A forward interest rate is the implied rate for a future period, derived from current spot rates. For example, the "1-year rate, 1 year forward" is the implied 1-year rate that will begin 1 year from today. Forward rates are calculated using the no-arbitrage principle that connects spot rates of different maturities.

Forward Rate Formula (Discrete Compounding)
(1 + r1)t1 * (1 + f)(t2-t1) = (1 + r2)t2
Solved: f = [(1 + r2)t2 / (1 + r1)t1]1/(t2-t1) - 1

The No-Arbitrage Principle

The forward rate relationship must hold to prevent arbitrage opportunities. If you invest $1 for t2 years at rate r2, you should get the same result as investing for t1 years at r1 and then reinvesting for (t2-t1) years at the forward rate f.

No-Arbitrage: If the forward rate relationship doesn't hold, traders could lock in risk-free profits by borrowing at one rate and investing at another.

Forward Rates and the Yield Curve

Upward-Sloping Curve

Forward rates exceed spot rates. Under the expectations hypothesis, this suggests markets anticipate rising rates—though term premiums can also explain higher forward rates. This is the most common yield curve shape.

Inverted Curve

Forward rates are below spot rates. This may signal expectations of falling rates, though other factors can contribute. Often observed before economic slowdowns.

Practical Applications

  • FRA Pricing: Forward Rate Agreements lock in forward rates for hedging
  • Swap Valuation: Forward rates are used to price interest rate swaps
  • Yield Curve Analysis: Forward rates reveal market expectations
  • Bond Portfolio Management: Riding the yield curve strategies
Important: Forward rates reflect market expectations but are not predictions. Actual future spot rates often differ from today's implied forward rates due to risk premiums and changing conditions.

Frequently Asked Questions

A spot rate is the interest rate for a loan or investment starting today, while a forward rate is the implied rate for a future period. Spot rates are directly observable in the market (e.g., Treasury yields), while forward rates are derived from the term structure of spot rates using no-arbitrage principles.

Forward rates are calculated using: f = [(1 + r2)t2 / (1 + r1)t1]1/(t2-t1) - 1. This formula ensures that investing at the spot rate for t2 years gives the same result as investing at the t1 spot rate and then reinvesting at the forward rate for the remaining period.

A negative forward rate indicates that the yield curve is inverted for that specific forward period—i.e., the longer-term spot rate (r2) is lower than the shorter-term rate (r1). This segment-level inversion doesn't necessarily mean the entire yield curve is inverted. Negative forward rates have been observed during periods of expected economic weakness or deeply accommodative monetary policy, though they remain relatively uncommon.

Forward Rate Agreements (FRAs) are derivative contracts that lock in a forward rate. The FRA rate should equal the implied forward rate from the spot curve. If they differ significantly, arbitrage opportunities exist. This calculator helps you determine the fair forward rate that should be used to price FRAs.

Forward rates reflect market expectations but are not reliable predictions of future spot rates. Research shows forward rates tend to overestimate future rates during normal times (due to term premiums). They're better viewed as risk-neutral expectations that incorporate both rate forecasts and risk compensation.

The forward rate relationship is fundamental to fixed income analysis because it: (1) ensures no-arbitrage conditions in the market, (2) helps price FRAs, swaps, and other derivatives, (3) reveals market expectations about future interest rates, and (4) is essential for yield curve construction and analysis in portfolio management.
Disclaimer

This calculator is for educational purposes only. Forward rates derived from this calculator assume discrete annual compounding. Actual market calculations may use different compounding conventions (e.g., continuous compounding, semi-annual) and day count methods. Forward rates are implied rates, not predictions of future rates. Consult a qualified professional for investment decisions.

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