Theforward rate is the future yield on abond. It is calculated using theyield curve. For example, the yield on a three-monthTreasury bill six months from now is aforward rate.^[1]

To extractthe forward rate, we need thezero-couponyield curve.

We are trying to find thefuture interest rate${\displaystyle r_{1,2}}$ for time period${\displaystyle (t_1,t_2)}$,${\displaystyle t_1}$ and${\displaystyle t_2}$ expressed inyears, given the rate${\displaystyle r_1}$ for time period${\displaystyle (0,t_1)}$ and rate${\displaystyle r_2}$ for time period${\displaystyle (0,t_2)}$. To do this, we use the property, following from thearbitrage-free pricing of bonds, that the proceeds from investing at rate${\displaystyle r_1}$ for time period${\displaystyle (0,t_1)}$ and thenreinvesting those proceeds at rate${\displaystyle r_{1,2}}$ for time period${\displaystyle (t_1,t_2)}$ is equal to the proceeds from investing at rate${\displaystyle r_2}$ for time period${\displaystyle (0,t_2)}$.

${\displaystyle r_{1,2}}$ depends on the rate calculation mode (simple,yearly compounded orcontinuously compounded), which yields three different results.

Mathematically it reads as follows:

${\displaystyle (1+r_1{t}_1)(1+r_{1,2}(t_2-t_1))=1+r_2{t}_2}$

Solving for${\displaystyle r_{1,2}}$ yields:

Thus${\displaystyle r_{1,2}=\frac{1}{t_2-t_1}\left(\frac{1+r_2{t}_2}{1+r_1{t}_1}-1\right)}$

The discount factor formula for period (0, t)${\displaystyle {\mathrm{\Delta }}_t}$ expressed in years, and rate${\displaystyle r_t}$ for this period being${\displaystyle DF(0,t)=\frac{1}{(1+r_t{\mathrm{\Delta }}_t)}}$,the forward rate can be expressed in terms of discount factors:${\displaystyle r_{1,2}=\frac{1}{t_2-t_1}\left(\frac{DF(0,t_1)}{DF(0,t_2)}-1\right)}$

${\displaystyle (1+r_1{)}^{t_1}(1+r_{1,2}{)}^{t_2-t_1}=(1+r_2{)}^{t_2}}$

Solving for${\displaystyle r_{1,2}}$ yields :

${\displaystyle r_{1,2}={\left(\frac{(1+r_2{)}^{t_2}}{(1+r_1{)}^{t_1}}\right)}^{1/(t_2-t_1)}-1}$

The discount factor formula for period (0,t)${\displaystyle {\mathrm{\Delta }}_t}$ expressed in years, and rate${\displaystyle r_t}$ for this period being${\displaystyle DF(0,t)=\frac{1}{(1+r_t{)}^{{\mathrm{\Delta }}_t}}}$, the forward rate can be expressed in terms of discount factors:

${\displaystyle r_{1,2}={\left(\frac{DF(0,t_1)}{DF(0,t_2)}\right)}^{1/(t_2-t_1)}-1}$

${\displaystyle e^{r_2\cdot t_2}=e^{r_1\cdot t_1}\cdot e^{r_{1,2}\cdot \left(t_2-t_1\right)}}$

Solving for${\displaystyle r_{1,2}}$ yields:

STEP 1→${\displaystyle e^{r_2\cdot t_2}=e^{r_1\cdot t_1+r_{1,2}\cdot \left(t_2-t_1\right)}}$

STEP 2→${\displaystyle \ln \left(e^{r_2\cdot t_2}\right)=\ln \left(e^{r_1\cdot t_1+r_{1,2}\cdot \left(t_2-t_1\right)}\right)}$

STEP 3→${\displaystyle r_2\cdot t_2=r_1\cdot t_1+r_{1,2}\cdot \left(t_2-t_1\right)}$

STEP 4→${\displaystyle r_{1,2}\cdot \left(t_2-t_1\right)=r_2\cdot t_2-r_1\cdot t_1}$

STEP 5→${\displaystyle r_{1,2}=\frac{r_2\cdot t_2-r_1\cdot t_1}{t_2-t_1}}$

The discount factor formula for period (0,t)${\displaystyle {\mathrm{\Delta }}_t}$ expressed in years, and rate${\displaystyle r_t}$ for this period being${\displaystyle DF(0,t)=e^{-r_t{\mathrm{\Delta }}_t}}$,the forward rate can be expressed in terms of discount factors:

${\displaystyle r_{1,2}=\frac{\ln \left(DF\left(0,t_1\right)\right)-\ln \left(DF\left(0,t_2\right)\right)}{t_2-t_1}=\frac{-\ln \left(\frac{DF\left(0,t_2\right)}{DF\left(0,t_1\right)}\right)}{t_2-t_1}}$

${\displaystyle r_{1,2}}$ is the forward rate between time${\displaystyle t_1}$ and time${\displaystyle t_2}$,

${\displaystyle r_k}$ is the zero-coupon yield for the time period${\displaystyle (0,t_k)}$, (k = 1,2).

• Forward rate agreement
• Floating rate note

• Forward price
• Spot rate

• Financial economics
• Swaps (finance)
• Fixed income analysis
• Interest rates

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