Forward rate

Forward rate calculation

To extract the forward rate, we need the zero-coupon yield curve.

We are trying to find the future interest rate ${\displaystyle r_{1,2}}$ for time period ${\displaystyle (t_1,t_2)}$, ${\displaystyle t_1}$ and ${\displaystyle t_2}$ expressed in years, 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 that the proceeds from investing at rate ${\displaystyle r_1}$ for time period ${\displaystyle (0,t_1)}$ and then reinvesting 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 or continuously compounded), which yields three different results.

Mathematically it reads as follows:

Simple rate

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

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

Yearly compounded rate

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

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:

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

Continuously compounded rate

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

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).