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Milds 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.

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

$r_{1,2}={\bigg (}{\frac {DF(0,t_{1})}{DF(0,t_{2})}}{\bigg )}^{\frac {1}{t_{2}-t_{1}}}-1$ Continuously compounded rate

$e^{r_{1}t_{1}}e^{r_{1,2}(t_{2}-t_{1})}=e^{r_{2}t_{2}}$ Solving for $r_{1,2}$ yields : $r_{1,2}={\frac {r_{2}t_{2}-r_{1}t_{1}}{t_{2}-t_{1}}}$ 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).