Finance:Forward rate

Short description: Future yield on a bond

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.

Forward rate calculation

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

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

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

Mathematically it reads as follows:

Simple rate

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

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

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

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

Yearly compounded rate

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

Solving for $r_{1,2}$ yields :

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

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

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

Continuously compounded rate

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

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

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

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

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

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

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

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

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

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

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

Related instruments

• Forward rate agreement
• Floating rate note

See also

• Forward price
• Spot rate