# 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:

$$ \left(1+s_n\right)^n=(1+f_1)\times\ (1+f_2)\times(1+f_3)\times...\times(1+f_n) $$

## 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:

$$ f_n=\frac{{(1+s_n)}^n}{{1+s_{n-1})}^{n-1}}-1 $$

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:

$$ {\rm FV}_1=$910\times{(1+4.5\%)}^2 $$

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:

$$ {\rm FV}_2=$910\times{(1+4.7\%)}^1\times(1+f_2) $$

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

$$ $910\times{(1+4.5\%)}^2=$910\times{(1+4.7\%)}^1\times(1+f_2) $$

Rearranging we get:

$$ f_2)=\frac{$910\times{(1+4.5\%)}^2}{$910\times{(1+4.7\%)}^1}-1 $$

Solving the above equation, f_{2} equals 4.3%.

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