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Annual Forward Rate from Two Spot Rates

Derive an annually compounded forward rate between two positive maturities from their zero-coupon spot rates.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Use the result with context

Results are estimates for planning and education. Confirm rates, taxes, fees, and legal requirements with the relevant institution or a qualified professional before making a financial decision.

Formula and method

A two-year spot rate of 10 percent and one-year spot rate of 0 percent imply a second-year forward rate of 21 percent.

(1 + S₁)^n₁ = (1 + S₂)^n₂ × (1 + f)^(n₁ − n₂)

Worked example

Enter these known values and leave the other values blank.

Later maturity from today
2 yrs
Annual spot rate to later maturity
10 %
Earlier maturity from today
1 yrs
Annual spot rate to earlier maturity
0 %
Annual forward rate between maturities
21 %

Assumptions and limitations

  • Rates are annual effective rates for zero-coupon accumulation, not coupon rates or continuously compounded yields. The earlier maturity is positive and strictly before the later maturity.
  • Rates can be negative but must exceed minus 100 percent so every annual accumulation factor remains positive.
  • The forward rate is implied by the entered curve under the accumulation identity; it is not a prediction of a future observed interest rate.
  • Maturity selectors use fixed month/year conversions. Equal rates can leave an inverse maturity undetermined even when the forward rate is well defined.

Common questions

Can the implied forward rate be negative?

Yes, provided its annual accumulation factor is positive. A downward spot curve can imply a negative forward rate.

Is this a forecast?

No. It is the rate that makes the two entered spot-rate accumulation paths agree.

References

Bookify requires each annual spot or forward accumulation factor to be positive, allowing rates above minus 100 percent.

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