The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a forward rate.

To extract the forward rate, we need the zero-coupon yield curve.

We are trying to find the future interest rate ${\displaystyle r_{1,2}}$ for time period ${\displaystyle (t_{1},t_{2})}$, ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ expressed in years, given the rate ${\displaystyle r_{1}}$ for time period ${\displaystyle (0,t_{1})}$ and rate ${\displaystyle r_{2}}$ for time period ${\displaystyle (0,t_{2})}$. To do this, we use the property, following from the arbitrage-free pricing of bonds, that the proceeds from investing at rate ${\displaystyle r_{1}}$ for time period ${\displaystyle (0,t_{1})}$ and then reinvesting those proceeds at rate ${\displaystyle r_{1,2}}$ for time period ${\displaystyle (t_{1},t_{2})}$ is equal to the proceeds from investing at rate ${\displaystyle r_{2}}$ for time period ${\displaystyle (0,t_{2})}$.

${\displaystyle r_{1,2}}$ depends on the rate calculation mode (simple, yearly compounded or continuously compounded), which yields three different results.

Mathematically it reads as follows:

Solving for ${\displaystyle r_{1,2}}$ yields:

Thus ${\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {1+r_{2}t_{2}}{1+r_{1}t_{1}}}-1\right)}$

The discount factor formula for period (0, t) ${\displaystyle \Delta _{t}}$ expressed in years, and rate ${\displaystyle r_{t}}$ for this period being ${\displaystyle DF(0,t)={\frac {1}{(1+r_{t}\,\Delta _{t})}}}$, the forward rate can be expressed in terms of discount factors: ${\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {DF(0,t_{1})}{DF(0,t_{2})}}-1\right)}$