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.^[1]

Forward rate calculation

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 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:

Simple rate

${\displaystyle (1+r_{1}t_{1})(1+r_{1,2}(t_{2}-t_{1}))=1+r_{2}t_{2))$

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: ${\displaystyle r_{1,2}={\frac {1}{t_{2}-t_{1))}\left({\frac {DF(0,t_{1})}{DF(0,t_{2})))-1\right)}$

Yearly compounded rate

${\displaystyle (1+r_{1})^{t_{1))(1+r_{1,2})^{t_{2}-t_{1))=(1+r_{2})^{t_{2))}$

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

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

The discount factor formula for period (0,t) ${\displaystyle \Delta _{t))$ 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:

${\displaystyle r_{1,2}=\left({\frac {DF(0,t_{1})}{DF(0,t_{2})))\right)^{1/(t_{2}-t_{1})}-1}$

Continuously compounded rate

${\displaystyle e^{r_{2}\cdot t_{2))=e^{r_{1}\cdot t_{1))\cdot \ e^{r_{1,2}\cdot \left(t_{2}-t_{1}\right)))$

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

STEP 1→ ${\displaystyle e^{r_{2}\cdot t_{2))=e^{r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)))$

STEP 2→ ${\displaystyle \ln \left(e^{r_{2}\cdot t_{2))\right)=\ln \left(e^{r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}\right)}$

STEP 3→ ${\displaystyle r_{2}\cdot t_{2}=r_{1}\cdot t_{1}+r_{1,2}\cdot \left(t_{2}-t_{1}\right)}$

STEP 4→ ${\displaystyle r_{1,2}\cdot \left(t_{2}-t_{1}\right)=r_{2}\cdot t_{2}-r_{1}\cdot t_{1))$

STEP 5→ ${\displaystyle r_{1,2}={\frac {r_{2}\cdot t_{2}-r_{1}\cdot t_{1)){t_{2}-t_{1))))$

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

${\displaystyle r_{1,2}={\frac {\ln \left(DF\left(0,t_{1}\right)\right)-\ln \left(DF\left(0,t_{2}\right)\right)}{t_{2}-t_{1))}={\frac {-\ln \left({\frac {DF\left(0,t_{2}\right)}{DF\left(0,t_{1}\right)))\right)}{t_{2}-t_{1))))$

${\displaystyle r_{1,2))$ is the forward rate between time ${\displaystyle t_{1))$ and time ${\displaystyle t_{2))$,

${\displaystyle r_{k))$ is the zero-coupon yield for the time period ${\displaystyle (0,t_{k})}$, (k = 1,2).

Related instruments

• Forward rate agreement
• Floating rate note