Forward rate
# 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

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

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

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

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