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.

Forward rate calculation

To extract the forward rate, we need the zero-coupon yield curve.

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:

Simple rate

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

Solving for $r_{t_{1},t_{2}}$ yields: $r_{t_{1},t_{2}}={\frac {1}{d_{2}-d_{1}}}\left({\frac {1+r_{2}d_{2}}{1+r_{1}d_{1}}}-1\right)$

Yearly compounded rate

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

Solving for $r_{t_{1},t_{2}}$ yields : $r_{t_{1},t_{2}}=\left({\frac {(1+r_{2})^{d_{2}}}{(1+r_{1})^{d_{1}}}}\right)^{\frac {1}{d_{2}-d_{1}}}-1$

Also, the discount factor formula for period (0, t) and rate $r_{t}$ being: $DF(0,t)={\frac {1}{(1+r_{t})^{t}}}$, the forward rate can be expressed in terms of discount factors: $r_{t_{1},t_{2}}={\bigg (}{\frac {DF(t_{1})}{DF(t_{2})}}{\bigg )}^{\frac {1}{t_{2}-t_{1}}}-1$

Continuously compounded rate

$e^{r_{1}d_{1}}e^{r_{t_{1},t_{2}}(d_{2}-d_{1})}=e^{r_{2}d_{2}}$

Solving for $r_{t_{1},t_{2}}$ yields : $r_{t_{1},t_{2}}={\frac {r_{2}d_{2}-r_{1}d_{1}}{d_{2}-d_{1}}}$

$r_{t_{1},t_{2}}$ 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 $(0,t_{2})$,

Related instruments

• Forward rate agreement
• Floating rate note

See also

• Forward price