Forward Interest Rate

Forward interest rate is the interest rate that can be locked today for some future period. It is the rate at which a party commits to borrow or lend a sum of money at some future date. Forward rates can be computed from spot interest rates (i.e. yields on zero-coupon bonds) through a process called bootstrapping.

Forward interest rates can be guaranteed through derivative contracts i.e. interest rate forward contracts (also called forward rate agreements), etc.

A spot interest rate for period with n years represents the cumulative effect of interest expectations of all n periods. It can be reconstructed as equal to the combined effect of the forward interest rate for first, second, third and nth period. This concept can be expressed mathematically as follows:

$$ \left(\text{1}+\text{s} _ \text{n}\right)^\text{n}=(\text{1}+\text{f} _ \text{1})\times\ (\text{1}+\text{f} _ \text{2})\times(\text{1}+\text{f} _ \text{3})\times\text{...}\times(\text{1}+\text{f} _ \text{n}) $$


From the equation above, it follows that the combined effect of n-1 forward rates for consecutive periods must equal the spot rate for n-1 periods.

Hence, it follows that the forward interest rate for period n in future can be determined using the following formula:

$$ \text{f} _ \text{n}=\frac{{(\text{1}+\text{s} _ \text{n})}^\text{n}}{{\text{1}+\text{s} _ {\text{n}-\text{1}})}^{\text{n}-\text{1}}}-\text{1} $$

Where fn is the future interest rate for period n in future, sn and sn-1 are the spot interest rates with n and n-1 years to maturity respectively. sn and sn-1 equal the yield to maturity on zero-coupon bonds with n and n-1 years till maturity respectively.


Let’s say you have $910 which you want to invest for next 2 years. You can invest them in a 2-year zero-coupon bond that yields 4.5% or you can investment it in a 1-year zero-coupon bond that yields 4.7% and simultaneously agree with a dealer to invest the proceeds received at the maturity of the 1-year zero-coupon bond with him at the end of first year at a forward rate f2.

You can determine the forward rate f2 that you should pay using the no-arbitrage principle which dictates that your total return must be equal in both scenarios.

If you invest $910 for 2 years at 4.5% your future value will be as follows:

$$ {\rm \text{FV}} _ \text{1}=\text{\$910}\times{(\text{1}+\text{4.5%})}^\text{2} $$

If you invest $910 at 4.7% for first year, the value of your investment will be $952.77 (=$910 × (1 + 4.7%)) which you have committed to invest at f2 for second year. Your investment value at the end of second year under this second scenario can be calculated as follows:

$$ {\rm \text{FV}} _ \text{2}=\text{\$910}\times{(\text{1}+\text{4.7%})}^\text{1}\times(\text{1}+\text{f} _ \text{2}) $$

Based on no-arbitrage principle, FV1 must equal FV2:

$$ \text{\$910}\times{(\text{1}+\text{4.5%})}^\text{2}=\text{\$910}\times{(\text{1}+\text{4.7%})}^\text{1}\times(\text{1}+\text{f} _ \text{2}) $$

Rearranging we get:

$$ \text{f} _ \text{2})=\frac{\text{\$910}\times{(\text{1}+\text{4.5%})}^\text{2}}{\text{\$910}\times{(\text{1}+\text{4.7%})}^\text{1}}-\text{1} $$

Solving the above equation, f2 equals 4.3%.

The above equation is the same as the formula given above.

by Obaidullah Jan, ACA, CFA and last modified on is a free educational website; of students, by students, and for students. You are welcome to learn a range of topics from accounting, economics, finance and more. We hope you like the work that has been done, and if you have any suggestions, your feedback is highly valuable. Let's connect!

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