# Forward Rate Calculator

Created by Wei Bin Loo
Reviewed by Hanna Pamuła, PhD and Jack Bowater
Based on research by
Tomas Björk, Bent Jesper Christensen Interest Rate Dynamics and Consistent Forward Rate Curves (April 21, 2004)
Last updated: Jan 15, 2023

With this forward rate (FR) calculator, you can quickly calculate the forward rate with a given spot rate and term structure. This calculator calculates the interest rate of an investment from the end of time period 2 to the end of time period 1. Our calculator focuses on calculating yearly compounded forward rates.

By reading this article, you will not only be able to understand what is forward rate and how to calculate forward rate, but you will also understand how to apply it in bond valuation.

## What is the forward rate?

A forward rate is the interest rate of an investment that will be initiated in the future. It is an estimation assuming that the market is perfectly efficient and no arbitrage opportunities exist. Forward rates can be determined using spot rates and the respective term structures.
Interest rates can also be adjusted by taking into account the predicted inflation. Our real interest rate calculator allows you to do it.

## How to calculate forward rate?

Calculating forward rate can be a complex concept to comprehend. So, allow us to explain it using an example.

As forward rates are most commonly used in valuing the future values of bonds, imagine you want to invest in the US corporate bond market for 5 years, there are 2 options that you can consider:

• Invest your money today in a 5-year US corporate bond; or
• Invest your money today in a 3-year US corporate bond then make a new investment into a 2-year US corporate bond using the proceeds for the previous 3-year investment.

In the first option, you can easily determine the yield on the 5-year outright investment; you can learn more about yields from our APY calculator. This yield is known as the spot rate. However, in the second option you only know the 3-year investment yield. Without calculating the forward rate, you would have no information on the 2-year bond yield on the investment that would happen 3 years from now. This yield that is not known on the investment made is the forward rate. In this example, the forward rate is the interest rate beginning 3 years from now, for a 2-year period.

To derive the forward rate, 4 inputs need to be found, as shown in the forward rate formula below:

$\scriptsize \left(\frac{(1 + S_1)^{n_1}}{(1 + S_2)^{n_2}} \right)^{\frac{1}{n_1 - n_2}} - 1$

Where,

• $n_1$ and $n_2$ - Time period 1 and 2, respectively. Time period 1 is the investment horizon of the longer investment. Hence, $n_1$ should be longer than $n_2$.

• $S_1$ - Spot rate for time period 1. It is the interest rate of an investment from today until time period 1, $n_1$.

• $S_2$ - Spot rate for time period 2. It is the interest rate of an investment from today until time period 2, $n_2$.

In our example, the inputs are:

• $n_1 = 5 \ \text{years}$ as the first option has an investment horizon of 5 years.
• $S_1 = 6\%$. This is the interest rate for the 5-year investment from today.
• $n_2 = 3 \ \text{years}$ as the future investment of the second option is 3 years away.
• $S_2 = 3\%$. This is the interest rate for the 3-year investment from today.

Hence, according to the forward rate formula, to calculate the forward rate:

\scriptsize \begin{aligned} \text{Forward rate} &= \left(\frac{(1 + 0.06)^5}{(1+0.03)^3}\right)^{\frac{1}{5-3}} - 1 \\ &= 10.66\% \end{aligned}

## Forward rates in practice

Using the example above, if you want to make a 5-year investment, you can directly invest in a 5-year US corporate bond or purchase a 3-year US corporate bond and roll it into a 2-year US corporate bond once it matures. In an efficient market where no arbitrage opportunity exists, the investor should be indifferent as both investments, in theory, should produce the same total return. If you're interested in returns on investments, check out the ROI calculator.

When you are making your investment, you can easily obtain the spot rates for the 5-year and 3-year investments. You, however, will not know the spot rate for the 2-year US corporate bond that will be purchased 3 years from now.

The second option hence presents reinvestment risks. But worry not, calculating forward rates can help you to solve this. You can choose to enter into a contractual agreement called a forward rate agreement to eliminate your reinvestment risk. The forward rate agreement is an agreement that will allow you to make an investment at the current forward rate that will take place in the future.

Now, fast-forward 1 year. If the spot rate for the new 2-year investment is lower, you could use the forward rate agreement to make the 2-year investment at agreed upon higher forward rate. If the spot rate is higher than the forward rate, the you will have the option to cancel the forward rate agreement and make a new 2-year investment based on the current spot rate.

But again, do keep in mind that the forward rate is merely an estimation in a perfectly efficient market with no friction. However, in real life, the market is inefficient, and some friction forces such as transaction costs still exist. Although there is no way to go around it, understand this concept can help you to understand how a market SHOULD work.

Wei Bin Loo
Time period 1
yrs
Spot rate for time period 1
%
Time period 2
yrs
Spot rate for time period 2
%
Forward rate
%
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