forward rate

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:

: $(1+r\_1t\_1)(1+\; r\_\{1,2\}(t\_2-t\_1))\; =\; 1+r\_2t\_2$

Solving for $r\_\{1,2\}$ yields:

Thus $r\_\{1,2\}\; =\; \backslash frac\{1\}\{t\_2-t\_1\}\backslash left(\backslash frac\{1+r\_2t\_2\}\{1+r\_1t\_1\}-1\backslash right)$

The discount factor formula for period (0, t) $\backslash Delta\_t$ expressed in years, and rate $r\_t$ for this period being

$DF(0,\; t)=\backslash frac\{1\}\{(1+r\_t\; \backslash ,\; \backslash Delta\_t)\}$,

the forward rate can be expressed in terms of discount factors:

$r\_\{1,2\}\; =\; \backslash frac\{1\}\{t\_2-t\_1\}\backslash left(\backslash frac\{DF(0,\; t\_1)\}\{DF(0,\; t\_2)\}-1\backslash right)$

: $(1+r\_1)^\{t\_1\}(1+r\_\{1,2\})^\{t\_2-t\_1\}\; =\; (1+r\_2)^\{t\_2\}$

Solving for $r\_\{1,2\}$ yields :

: $r\_\{1,2\}\; =\; \backslash left(\backslash frac\{(1+r\_2)^\{t\_2\}\}\{(1+r\_1)^\{t\_1\}\}\backslash right)^\{1/(t\_2-t\_1)\}\; -\; 1$

The discount factor formula for period (0,t) $\backslash Delta\_t$ expressed in years, and rate $r\_t$ for this period being

$DF(0,\; t)=\backslash frac\{1\}\{(1+r\_t)^\{\backslash Delta\_t\}\}$, the forward rate can be expressed in terms of discount factors:

: $r\_\{1,2\}=\backslash left(\backslash frac\{DF(0,\; t\_1)\}\{DF(0,\; t\_2)\}\backslash right)^\{1/(t\_2-t\_1)\}-1$

:$e^\{r\_2\; \backslash cdot\; t\_2\}\; =\; e^\{r\_1\; \backslash cdot\; t\_1\}\; \backslash cdot\; \backslash \; e^\{r\_\{1,2\}\; \backslash cdot\; \backslash left(t\_2\; -\; t\_1\; \backslash right)\}$

Solving for $r\_\{1,2\}$ yields:

:STEP 1→ $e^\{r\_2\; \backslash cdot\; t\_2\}\; =\; e^\{r\_1\; \backslash cdot\; t\_1\; +\; r\_\{1,2\}\; \backslash cdot\; \backslash left(t\_2\; -\; t\_1\; \backslash right)\}$

:STEP 2→ $\backslash ln\; \backslash left(e^\{r\_2\; \backslash cdot\; t\_2\}\; \backslash right)\; =\; \backslash ln\; \backslash left(e^\{r\_1\; \backslash cdot\; t\_1\; +\; r\_\{1,2\}\; \backslash cdot\; \backslash left(t\_2\; -\; t\_1\; \backslash right)\}\backslash right)$

:STEP 3→ $r\_2\; \backslash cdot\; t\_2\; =\; r\_1\; \backslash cdot\; t\_1\; +\; r\_\{1,2\}\; \backslash cdot\; \backslash left(t\_2\; -\; t\_1\; \backslash right)$

:STEP 4→ $r\_\{1,2\}\; \backslash cdot\; \backslash left(t\_2\; -\; t\_1\; \backslash right)\; =\; r\_2\; \backslash cdot\; t\_2\; -\; r\_1\; \backslash cdot\; t\_1$

:STEP 5→ $r\_\{1,2\}\; =\; \backslash frac\{\; r\_2\; \backslash cdot\; t\_2\; -\; r\_1\; \backslash cdot\; t\_1\}\{t\_2\; -\; t\_1\}$

The discount factor formula for period (0,t) $\backslash Delta\_t$ expressed in years, and rate $r\_t$ for this period being

$DF(0,\; t)=e^\{-r\_t\backslash ,\backslash Delta\_t\}$,

the forward rate can be expressed in terms of discount factors:

: $r\_\{1,2\}\; =\; \backslash frac\{\backslash ln\; \backslash left(DF\; \backslash left(0,\; t\_1\; \backslash right)\backslash right)\; -\; \backslash ln\; \backslash left(DF\; \backslash left(0,\; t\_2\; \backslash right)\backslash right)\}\{t\_2\; -\; t\_1\}$

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

$r\_\{1,2\}$ 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).

• Forward rate agreement
• Floating rate note

• Forward price
• Spot rate

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Category:Financial economics

Category:Swaps (finance)

Category:Fixed income analysis

Category:Interest rates