Editing Forward rate (section)

===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).