# Forward rate

Not to be confused with forward price or forward exchange 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]

## Forward rate calculation

To extract the forward rate, one needs the zero-coupon yield curve. The general formula used to calculate the forward rate is:

### Simple rate

$r_{t_1,t_2} = \frac{1}{d_2-d_1}\left(\frac{1+r_2d_2}{1+r_1d_1}-1\right)$

### Compound rate

$r_{t_1,t_2} = \left(\frac{(1+r_2)^{d_2}}{(1+r_1)^{d_1}}\right)^{\frac{1}{d_2-d_1}} - 1$

### Exponential rate

$r_{t_1,t_2} = \frac{r_2d_2-r_1d_1}{d_2-d_1}$

$r_{t_1,t_2}$ is the forward rate between term $t_1$ and term $t_2$,

$d_1$ is the time length between time 0 and term $t_1$ (in years),

$d_2$ is the time length between time 0 and term $t_2$ (in years),

$r_1$ is the zero-coupon yield for the time period $(0, t_1)$,

$r_2$ is the zero-coupon yield for the time period $(0, t_2)$,

### Derivation

We are trying to find the future interest rate for time period $(t_1, t_2)$, 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 solve for the interest rate $r_{t_1,t_2}$ for time period $(t_1, t_2)$ for which the proceeds from investing at rate $r_1$ for time period $(0, t_1)$ and then reinvesting those proceeds at rate $r_{t_1,t_2}$ for time period $(t_1, t_2)$ is equal to the proceeds from investing at rate $r_2$ for time period $(0, t_2)$. Or, mathematically:

$(1+r_1)^{d_1}(1+r_{t_1,t_2})^{d_2-d_1} = (1+r_2)^{d_2}$

Solving for $r_{t_1,t_2}$ yields the above formula.

## Related instruments

A forward discount is when the forward rate of one currency relative to another currency is higher than the spot rate.

A forward premium is when the forward rate of one currency relative to another currency is lower than the spot rate.