Template:Short description Template:Distinguish 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.<ref>Template:Citation.</ref>
Forward rate calculationEdit
To extract the forward rate, we need the 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 rateEdit
- <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 rateEdit
- <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 rateEdit
- <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).