# 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:

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

## Formula

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 f_{n} is the future interest rate for period n in future, sn and s_{n-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.

## Example

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 f_{2}.

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, f_{2} equals 4.3%.

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

Written by Obaidullah Jan, ACA, CFA and last modified on