Forward rate

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

 * $$(1+r_1t_1)(1+ r_{1,2}(t_2-t_1)) = 1+r_2t_2$$

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

Thus $$r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{1+r_2t_2}{1+r_1t_1}-1\right)$$ The discount factor formula for period (0, t) $$\Delta_t$$ expressed in years, and rate $$r_t$$ for this period being $$DF(0, t)=\frac{1}{(1+r_t \, \Delta_t)}$$, the forward rate can be expressed in terms of discount factors: $$r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{DF(0, t_1)}{DF(0, t_2)}-1\right)$$

Yearly compounded rate

 * $$(1+r_1)^{t_1}(1+r_{1,2})^{t_2-t_1} = (1+r_2)^{t_2}$$

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


 * $$r_{1,2} = \left(\frac{(1+r_2)^{t_2}}{(1+r_1)^{t_1}}\right)^{1/(t_2-t_1)} - 1$$

The discount factor formula for period (0,t) $$\Delta_t$$ expressed in years, and rate $$r_t$$ for this period being $$DF(0, t)=\frac{1}{(1+r_t)^{\Delta_t}}$$, the forward rate can be expressed in terms of discount factors:


 * $$r_{1,2}=\left(\frac{DF(0, t_1)}{DF(0, t_2)}\right)^{1/(t_2-t_1)}-1$$

Continuously compounded rate

 * $$e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1} \cdot \ e^{r_{1,2} \cdot \left(t_2 - t_1 \right)}$$

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


 * STEP 1→  $$e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)}$$


 * STEP 2→  $$\ln \left(e^{r_2 \cdot t_2} \right) = \ln \left(e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)}\right)$$


 * STEP 3→  $$r_2 \cdot t_2 = r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)$$


 * STEP 4→  $$r_{1,2} \cdot \left(t_2 - t_1 \right) = r_2 \cdot t_2 - r_1 \cdot t_1$$


 * STEP 5→  $$r_{1,2} = \frac{ r_2 \cdot t_2 - r_1 \cdot t_1}{t_2 - t_1}$$

The discount factor formula for period (0,t) $$\Delta_t$$ expressed in years, and rate $$r_t$$ for this period being $$DF(0, t)=e^{-r_t\,\Delta_t}$$, the forward rate can be expressed in terms of discount factors:


 * $$r_{1,2} = \frac{\ln \left(DF \left(0, t_1 \right)\right) - \ln \left(DF \left(0, t_2 \right)\right)}{t_2 - t_1}

= \frac{- \ln \left( \frac{ DF \left(0, t_2 \right)}{ DF \left(0, t_1 \right)} \right)}{t_2 - t_1} $$

$$r_{1,2} $$ 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

 * Forward rate agreement
 * Floating rate note