Forward Rate Calculator

Find the implied forward interest rate from two spot rates, or solve for either spot rate or maturity. For yield curve, bond, and swap pricing work.

https://hexacalculator.com/calculators/finance/bonds/forward-rate-calculator

Finance

Bonds

Forward Rate Calculator

Find the implied forward interest rate from two spot rates, or solve for either spot rate or maturity. For yield curve, bond, and swap pricing work.

Forward Rate Calculator

years

%

years

%
%

Use continuous compounding

Switch the whole calculation from annual to continuously compounded rates.

Enter the two maturities and their spot rates, and leave the forward rate blank to solve for it. You can also leave any single field blank and fill the other four to solve for that one instead.

Please input values into the fields

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Show the no-arbitrage check

Compare the two equivalent investment paths side by side.

Show the spot-vs-forward chart

Plot the two spot rates and the implied forward on the maturity axis.

The forward rate is the interest rate that the market calculates backwards for a particular period in the future from today's spot rates. It cannot be directly shown on screen like yields of 1-year or 2-year bonds, but it is already determined by the available interest rates.

Forward Rate - Introduction
Forward Rate - Introduction

This calculator allows you to calculate either the forward rate itself, one of the spot rates, or any maturity.

What is a forward rate?

The spot rate is the yield on zero-coupon bonds of different maturities at a given point in time today. The forward rate is between two maturities and gives the interest rate that will apply for the period between those two maturities. If the spot rates are known for 1 year and 2 years, then the market has already determined the 1-year forward rate starting one year from now. This is because other values would create arbitrage opportunities.

Forward Rate - Explainer
Forward Rate - Explainer

The logic is based entirely on the principle of arbitrage-free pricing. If you invest your money at a long-term spot rate, then the end result should be exactly the same as if you invested it first at a short-term rate and then reinvested for the remaining time at the forward rate. When the returns from these two paths are different, traders will borrow cheaply and lend expensively until the difference disappears.

Understanding of formula.

Let n1 be the longer maturity with spot rate s1 and n2 be the shorter maturity with spot rate s2. The forward rate f is for the period from n2 to n1. By the principle of no arbitrage, these values are related as follows:

(1+s1)n1=(1+s2)n2 (1+f) n1−n2(1 + s_1)^{n_1} = (1 + s_2)^{n_2}\,(1 + f)^{\,n_1 - n_2}
Forward Rate - Formula
Forward Rate - Formula

A summary of forward rates leads to a simple closed-form formula.

f=[(1+s1)n1(1+s2)n2]1n1−n2−1f = \left[\frac{(1 + s_1)^{n_1}}{(1 + s_2)^{n_2}}\right]^{\frac{1}{n_1 - n_2}} - 1

Assume the spot rate for a 2-year maturity is 4 percent and the spot rate for 1-year is 3 percent.

Forward Rate - Example
Forward Rate - Example

The implied forward rate for 1-year beginning in 1 year would be as follows:

f=[(1.04)2(1.03)1]11−1≈0.0501f = \left[\frac{(1.04)^{2}}{(1.03)^{1}}\right]^{\frac{1}{1}} - 1 \approx 0.0501
Forward Rate - Calculation
Forward Rate - Calculation

Thus, the one-year interest rate implied in the market for a maturity of 1 year is approximately 5.01 percent. The following table shows the individual input values and examples.

Symbol

Meaning

Example

n2

Shorter maturity

1 year

s2

Spot rate at the shorter maturity

3 percent

n1

Longer maturity

2 years

s1

Spot rate at the longer maturity

4 percent

f

Forward rate from n2 to n1

result

The Difference Between Forward and Spot Rates

Spot rates apply for a specific period of time up to a particular maturity date starting today. Forward rates apply for the period between two future dates neither of which is today. Both are not mutually competing interest rates. Forward rates consist entirely of spot rates and only exist if the spot rate curve has a certain shape. If the spot rate curve is rising, forward rates will be higher than spot rates. If the curve is falling, forward rates will be lower than spot rates.

Forward Rate - Explainer
Forward Rate - Explainer

Annual compounding and continuous compounding.

The default setting uses annual compounding. If continuous compounding is enabled, the ratio between the two values can be expressed in a simple form as the weighted difference of interest rates.

f=s1n1−s2n2n1−n2f = \frac{s_1 n_1 - s_2 n_2}{n_1 - n_2}

Although the results obtained with both methods are similar, they are not identical. Therefore, select the appropriate settings for calculation tools according to how current interest rates are quoted. Derivatives trading desks usually use continuous time. In contrast, bond yields offered often apply discrete compounding.

Use of forward rates

Forward rates are essential data required for pricing of all products whose value depends on the future shape of the curve. In bond trading departments they are used to price fixed income securities and in interest rate derivatives market making, discounting is performed using forward curves.

Forward Rate - Explainer
Forward Rate - Explainer

Portfolio managers view the forward curve as a market-implied path for interest rates and trade in areas that deviate from their own expectations. A steeply rising forward curve indicates that the market expects future rate hikes or strong growth, while an inverted curve suggests cuts to rates, a slowdown, or both.

Tips for accurate results:

Please ensure that the longer rate's maturity strictly exceeds the shorter rate's maturity. The forward rate fills in the period between them so there is no desired value when maturities are equal and reversing the order of maturities changes the sign.

Take two spot interest rates from the same curve. In the USD market, yields on US government bonds are normally used. For other currencies there are also government bond curves. Mixing US yields with swap rates or corporate bond yields leads to forward values that do not reflect a real transaction.

Frequently asked questions

Can forward rates be negative?

Yes they can. If the spot rate for a longer maturity is significantly lower than the spot rate for a shorter maturity then the formula will result in a negative forward rate. European and Japanese government bond markets have had negative forward rates over several years while central banks have been operating with negative policy rates.

What is the difference between a forward rate and a futures rate?

Forward rates are calculated from the current spot yield curve. In contrast, futures rates come from traded contracts such as SOFR or Euribor futures. Although both move, they are not identical. This is because futures are re-priced daily and small adjustments are made that are not included in forward rates.

Are forward rates predictions of future interest rates?

No, it is not an economic science prediction. It's the interest rates that must hold today to rule out arbitrage opportunities between different maturities. The realized interest rates often deviate from previous forward rates, and as studies by Fama and Bliss have already shown, forward rates are a biased predictor because they contain both a futures component and a risk premium.

Why does the longer tenor have to be higher than the shorter one?

Since the forward rate reflects the period between two maturities, this period must be positive. If the maturities are equal there is no period over which to price and reversing the order of the maturities results in a nonsensical negative length.

What time unit should be used?

Both maturities must be expressed in years, with some years being represented as decimals. So 18 months would be 1.5 years. Since spot rates are quoted as annual interest rates, expressing the maturity uniformly in years can ensure consistency when compounding interest.

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Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

  1. Investopedia: Forward Rate

    Definition and worked calculation of a forward rate from spot rates.

  2. Investopedia: Term Structure of Interest Rates

    How the spot curve and its slope drive implied forward rates.