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 $r_$ 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 $r_$ for time period $(t_1, t_2)$ is equal to the proceeds from investing at rate $r_2$ for time period $(0, t_2)$.

$r_$ depends on the rate calculation mode (simple, yearly compounded or continuously compounded), which yields three different results.

Mathematically it reads as follows:

Simple rate

: $(1+r_1t_1)(1+ r_(t_2-t_1)) = 1+r_2t_2$

Solving for $r_$ yields:

Thus $r_ = \frac\left(\frac-1\right)$

The discount factor formula for period (0, t) $\Delta_t$ expressed in years, and rate $r_t$ for this period being
$DF(0, t)=\frac$,
the forward rate can be expressed in terms of discount factors:
$r_ = \frac\left(\frac-1\right)$

Yearly compounded rate

: $(1+r_1)^(1+r_)^ = (1+r_2)^$

Solving for $r_$ yields :

: $r_ = \left(\frac\right)^ - 1$

The discount factor formula for period (0,''t'') $\Delta_t$ expressed in years, and rate $r_t$ for this period being
$DF(0, t)=\frac$, the forward rate can be expressed in terms of discount factors:

: $r_=\left(\frac\right)^-1$

Continuously compounded rate

:$e^ = e^ \cdot \ e^$

Solving for $r_$ yields:

:STEP 1→ $e^ = e^$

:STEP 2→ $\ln \left(e^ \right) = \ln \left(e^\right)$

:STEP 3→ $r_2 \cdot t_2 = r_1 \cdot t_1 + r_ \cdot \left(t_2 - t_1 \right)$

:STEP 4→ $r_ \cdot \left(t_2 - t_1 \right) = r_2 \cdot t_2 - r_1 \cdot t_1$

:STEP 5→ $r_ = \frac$

The discount factor formula for period (0,''t'') $\Delta_t$ expressed in years, and rate $r_t$ for this period being
$DF(0, t)=e^$,
the forward rate can be expressed in terms of discount factors:

: $r_ = \frac = \frac$

$r_$ 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

See also

*Forward price
*Spot rate

References

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Financial economics
Swaps (finance)
Fixed income analysis
Interest rates