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.^[1]

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:

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

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

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

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:

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

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

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).

• Forward rate agreement
• Floating rate note

• Forward price
• Spot rate