# Forward rate

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.

### 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 $r_{1,2}={\frac {1}{t_{2}-t_{1}}}\left({\frac {1+r_{2}t_{2}}{1+r_{1}t_{1}}}-1\right)$

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

### 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) $\Delta _{t}$  expressed in years, and rate $r_{t}$  for this period being $DF(0,t)={\frac {1}{(1+r_{t})^{\Delta _{t}}}}$ , 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

EQUATION→ $e^{{(r}_{2}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})}\ast \ e^{\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}$

Solving for $r_{1,2}$  yields:

STEP 1→ $e^{{(r}_{2}\ast t_{2})}=e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}$

STEP 2→ $\ln {\left(e^{{(r}_{2}\ast t_{2})}\right)}=\ln {\left(e^{{(r}_{1}\ast t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)}\right)}$

STEP 3→ ${(r}_{2}\ast \ t_{2})={(r}_{1}\ast \ t_{1})+\left(r_{1,2}\ast \left(t_{2}-t_{1}\right)\right)$

STEP 4→ $r_{1,2}\ast \left(t_{2}-t_{1}\right)={(r}_{2}\ast \ t_{2})-{(r}_{1}\ast \ t_{1})$

STEP 5→ $r_{1,2}={\frac {{(r}_{2}\ast t_{2})-{(r}_{1}\ast t_{1})}{t_{2}-t_{1}}}$

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

$r_{1,2}={\frac {1}{t_{2}-t_{1}}}(\ln DF(0,t_{1})-\ln DF(0,t_{2}))$

$r_{1,2}$  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).