Editing Forward rate

{{Short description|Future yield on a bond}}
{{distinguish|forward price|forward exchange rate}}
The '''forward rate''' is the future yield on a [[bond (finance)|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''.<ref>{{Citation |last=Fabozzi |first=Vamsi.K|title=The Handbook of Fixed Income Securities |edition=Seventh |location=New York |publisher=kvrv |year=2012 |isbn=978-0-07-144099-8 |page=148 }}.</ref>

==Forward rate calculation==

To extract the forward rate, we need the [[Zero-coupon bond|zero-coupon]] [[yield curve]]. 

We are trying to find the future interest rate <math>r_{1,2}</math> for time period <math>(t_1, t_2)</math>, <math>t_1</math> and <math>t_2</math> expressed in '''years''', given the rate <math>r_1</math> for time period <math>(0, t_1)</math> and rate <math>r_2</math> for time period <math>(0, t_2)</math>. To do this, we use the property that the proceeds from investing at rate <math>r_1</math> for time period <math>(0, t_1)</math> and then '''reinvesting''' those proceeds at rate <math>r_{1,2}</math> for time period <math>(t_1, t_2)</math> is equal to the proceeds from investing at rate <math>r_2</math> for time period <math>(0, t_2)</math>.

<math>r_{1,2}</math> depends on the rate calculation mode ('''simple''', '''yearly compounded''' or '''continuously compounded'''), which yields three different results.

Mathematically it reads as follows:

===Simple rate===

: <math>(1+r_1t_1)(1+ r_{1,2}(t_2-t_1)) = 1+r_2t_2</math>
 		 		
Solving for <math>r_{1,2}</math> yields:

Thus <math>r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{1+r_2t_2}{1+r_1t_1}-1\right)</math>
 	
The discount factor formula for period (0, t) <math>\Delta_t</math> expressed in years, and rate <math>r_t</math> for this period being 	
<math>DF(0, t)=\frac{1}{(1+r_t \, \Delta_t)}</math>,
the forward rate can be expressed in terms of discount factors:	
<math>r_{1,2} = \frac{1}{t_2-t_1}\left(\frac{DF(0, t_1)}{DF(0, t_2)}-1\right)</math>

===Yearly compounded rate===

: <math>(1+r_1)^{t_1}(1+r_{1,2})^{t_2-t_1} = (1+r_2)^{t_2}</math>

Solving for <math>r_{1,2}</math> yields :

: <math>r_{1,2} = \left(\frac{(1+r_2)^{t_2}}{(1+r_1)^{t_1}}\right)^{1/(t_2-t_1)} - 1</math>

The discount factor formula for period (0,''t'') <math>\Delta_t</math> expressed in years, and rate <math>r_t</math> for this period being
<math>DF(0, t)=\frac{1}{(1+r_t)^{\Delta_t}}</math>, the forward rate can be expressed in terms of discount factors:

: <math>r_{1,2}=\left(\frac{DF(0, t_1)}{DF(0, t_2)}\right)^{1/(t_2-t_1)}-1</math>

===Continuously compounded rate===

:<math>e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1} \cdot \ e^{r_{1,2} \cdot \left(t_2 - t_1 \right)}</math>


Solving for <math>r_{1,2}</math> yields:


:'''STEP 1→'''   <math>e^{r_2 \cdot t_2} = e^{r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)}</math>

:'''STEP 2→'''   <math>\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)</math>

:'''STEP 3→'''   <math>r_2 \cdot t_2 = r_1 \cdot t_1 + r_{1,2} \cdot \left(t_2 - t_1 \right)</math>

:'''STEP 4→'''   <math>r_{1,2} \cdot \left(t_2 - t_1 \right) = r_2 \cdot t_2 - r_1 \cdot t_1</math>

:'''STEP 5→'''   <math>r_{1,2} = \frac{ r_2 \cdot t_2 - r_1 \cdot t_1}{t_2 - t_1}</math>

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

: <math>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} </math>

<math>r_{1,2} </math> is the forward rate between time <math> t_1 </math> and time <math> t_2 </math>, 

<math> r_k </math> is the zero-coupon yield for the time period <math> (0, t_k) </math>, (''k'' = 1,2).

== Related instruments ==
* [[Forward rate agreement]]
* [[Floating rate note]]

== See also ==
*[[Forward price]]
*[[Spot rate]]

== References ==
{{Reflist}}

[[Category:Financial economics]]
[[Category:Swaps (finance)]]
[[Category:Fixed income analysis]]
[[Category:Interest rates]]