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.

Forward rate calculation

To extract the forward rate, we need the zero-coupon yield curve. We are trying to find the future interest rate r_{1,2} for time period (t_1, t_2), t_1 and t_2 expressed in years, given the rate r_1 for time period (0, t_1) and rate r_2 for time period (0, t_2). To do this, we use the property that the proceeds from investing at rate r_1 for time period (0, t_1) and then reinvesting those proceeds at rate r_{1,2} for time period (t_1, t_2) is equal to the proceeds from investing at rate r_2 for time period (0, t_2). 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:

Simple rate

Solving for r_{1,2} yields: Thus The discount factor formula for period (0, t) \Delta_t expressed in years, and rate r_t for this period being , the forward rate can be expressed in terms of discount factors:

Yearly compounded rate

Solving for r_{1,2} yields : The discount factor formula for period (0,t) \Delta_t expressed in years, and rate r_t for this period being , the forward rate can be expressed in terms of discount factors:

Continuously compounded rate

Solving for r_{1,2} yields: The discount factor formula for period (0,t) \Delta_t expressed in years, and rate r_t for this period being , the forward rate can be expressed in terms of discount factors: r_{1,2} is the forward rate between time t_1 and time t_2, r_k is the zero-coupon yield for the time period (0, t_k), (k = 1,2).

Related instruments