Future yield on a bond

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]}

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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:

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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).